state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have ex := exists_null_frontiers_thickening μ F | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | let rs := Classical.choose ex | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have rs_lim : Tendsto rs atTop (𝓝 0) := (Classical.choose_spec ex).1 | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have rs_pos : ∀ n, 0 < rs n := fun n ↦ ((Classical.choose_spec ex).2 n).1 | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have rs_null : ∀ n, μ (frontier (Metric.thickening (rs n) F)) = 0 :=
fun n ↦ ((Classical.choose_spec ex).2 n).2 | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have Fthicks_open : ∀ n, IsOpen (Metric.thickening (rs n) F) :=
fun n ↦ Metric.isOpen_thickening | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have key := fun (n : ℕ) ↦ h (Fthicks_open n).measurableSet (rs_null n) | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply ENNReal.le_of_forall_pos_le_add | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | intros ε ε_pos μF_finite | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have keyB := tendsto_measure_cthickening_of_isClosed (μ := μ) (s := F)
⟨1, ⟨by simp only [gt_iff_lt, zero_lt_one], measure_ne_top _ _⟩⟩ F_closed | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [gt_iff_lt, zero_lt_one] | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have nhd : Iio (μ F + ε) ∈ 𝓝 (μ F) := by
apply Iio_mem_nhds
exact ENNReal.lt_add_right μF_finite.ne (ENNReal.coe_pos.mpr ε_pos).ne' | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply Iio_mem_nhds | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | exact ENNReal.lt_add_right μF_finite.ne (ENNReal.coe_pos.mpr ε_pos).ne' | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | specialize rs_lim (keyB nhd) | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [mem_map, mem_atTop_sets, ge_iff_le, mem_preimage, mem_Iio] at rs_lim | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | obtain ⟨m, hm⟩ := rs_lim | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.intro
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have aux' := fun i ↦ measure_mono (μ := μs i) (Metric.self_subset_thickening (rs_pos m) F) | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.intro
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have aux : (fun i ↦ (μs i F)) ≤ᶠ[L] (fun i ↦ μs i (Metric.thickening (rs m) F)) :=
eventually_of_forall aux' | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.intro
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | refine (limsup_le_limsup aux).trans ?_ | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.intro
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | rw [Tendsto.limsup_eq (key m)] | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.intro
Ω✝ : Type u_1
inst✝⁷ : PseudoEMetricSpace Ω✝
inst✝⁶ : MeasurableSpace Ω✝
inst✝⁵ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁴ : NeBot L
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply (measure_mono (Metric.thickening_subset_cthickening (rs m) F)).trans (hm m rfl.le).le | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib.MeasureTheory.Measure.Portmanteau.425_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁸ : PseudoEMetricSpace Ω✝
inst✝⁷ : MeasurableSpace Ω✝
inst✝⁶ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁵ : NeBot L
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply le_measure_liminf_of_limsup_measure_compl_le G_open.measurableSet | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all open sets G we have the limsup condition μ(G) ≤ liminf μsᵢ(G). -/
lemma le_limin... | Mathlib.MeasureTheory.Measure.Portmanteau.463_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all open sets G we have the limsup condition μ(G) ≤ liminf μsᵢ(G). -/
lemma le_limin... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω✝ : Type u_1
inst✝⁸ : PseudoEMetricSpace Ω✝
inst✝⁷ : MeasurableSpace Ω✝
inst✝⁶ : OpensMeasurableSpace Ω✝
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝⁵ : NeBot L
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | exact limsup_measure_closed_le_of_forall_tendsto_measure h _ (isClosed_compl_iff.mpr G_open) | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all open sets G we have the limsup condition μ(G) ≤ liminf μsᵢ(G). -/
lemma le_limin... | Mathlib.MeasureTheory.Measure.Portmanteau.463_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all open sets G we have the limsup condition μ(G) ≤ liminf μsᵢ(G). -/
lemma le_limin... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
f : Ω → ℝ
f_cont : Continuous f
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
⊢ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp_rw [lintegral_eq_lintegral_meas_lt _ (eventually_of_forall f_nn) f_cont.aemeasurable] | lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫⁻ x, ENNReal.ofReal ... | Mathlib.MeasureTheory.Measure.Portmanteau.497_0.COExymZuBXoOgED | lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫⁻ x, ENNReal.ofReal ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
f : Ω → ℝ
f_cont : Continuous f
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
⊢ ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t < f a} ≤ liminf (fun i => ∫⁻ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | calc ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a}
≤ ∫⁻ (t : ℝ) in Set.Ioi 0, atTop.liminf (fun i ↦ (μs i) {a | t < f a}) := ?_ -- (i)
_ ≤ atTop.liminf (fun i ↦ ∫⁻ (t : ℝ) in Set.Ioi 0, (μs i) {a | t < f a}) := ?_ | lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫⁻ x, ENNReal.ofReal ... | Mathlib.MeasureTheory.Measure.Portmanteau.497_0.COExymZuBXoOgED | lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫⁻ x, ENNReal.ofReal ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case calc_1
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
f : Ω → ℝ
f_cont : Continuous f
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
⊢ ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t < f a} ≤ ∫⁻ (t : ℝ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | exact (lintegral_mono (fun t ↦ h_opens _ (continuous_def.mp f_cont _ isOpen_Ioi))).trans
(le_refl _) | lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫⁻ x, ENNReal.ofReal ... | Mathlib.MeasureTheory.Measure.Portmanteau.497_0.COExymZuBXoOgED | lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫⁻ x, ENNReal.ofReal ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case calc_2
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
f : Ω → ℝ
f_cont : Continuous f
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
⊢ ∫⁻ (t : ℝ) in Ioi 0, liminf (fun i => ↑↑(μs i) {a ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | exact lintegral_liminf_le (fun n ↦ Antitone.measurable (fun s t hst ↦
measure_mono (fun ω hω ↦ lt_of_le_of_lt hst hω))) | lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫⁻ x, ENNReal.ofReal ... | Mathlib.MeasureTheory.Measure.Portmanteau.497_0.COExymZuBXoOgED | lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫⁻ x, ENNReal.ofReal ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
⊢ ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
f.continuous f_nn h_opens | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
f.continuous.measurable.aestronglyMeasurable] | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_4
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G)... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
f.continuous.measurable.aestronglyMeasurable] | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_4
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G)... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_4
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G)... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_4
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G)... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound) | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case convert_1
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i)... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | exact (f.lintegral_of_real_lt_top μ).ne | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case convert_2
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i)... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply ne_of_lt | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case convert_2.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case convert_2.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [measure_univ, mul_one] at obs | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case convert_2.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top) | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply liminf_le_of_le | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case hf
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atT... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | refine ⟨0, eventually_of_forall (by simp only [ge_iff_le, zero_le, forall_const])⟩ | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [ge_iff_le, zero_le, forall_const] | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTo... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | intro x hx | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTo... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | obtain ⟨i, hi⟩ := hx.exists | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.intro
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply le_trans hi | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.intro
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | convert obs i with x | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_3.h.e'_4.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have aux := ENNReal.ofReal_eq_coe_nnreal (f_nn x) | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_3.h.e'_4.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [ContinuousMap.toFun_eq_coe, BoundedContinuousFunction.coe_to_continuous_fun] at aux | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_3.h.e'_4.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | rw [aux] | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_3.h.e'_4.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | congr | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_3.h.e'_4.h.e_a.e_val
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (f... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | exact (Real.norm_of_nonneg (f_nn x)).symm | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib.MeasureTheory.Measure.Portmanteau.512_0.COExymZuBXoOgED | lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
{f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
(h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x : Ω), f x ∂μ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | rcases eq_or_neBot L with rfl|hL | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inl
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x : Ω),... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [tendsto_bot] | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x : Ω... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have obs := BoundedContinuousFunction.isBounded_range_integral μs f | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x : Ω... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have bdd_above : IsBoundedUnder (· ≤ ·) L (fun i ↦ ∫ (x : Ω), f x ∂μs i) :=
isBounded_le_map_of_bounded_range _ obs | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x : Ω... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have bdd_below : IsBoundedUnder (· ≥ ·) L (fun i ↦ ∫ (x : Ω), f x ∂μs i) :=
isBounded_ge_map_of_bounded_range _ obs | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x : Ω... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply @tendsto_of_le_liminf_of_limsup_le ℝ ι _ _ _ L (fun i ↦ ∫ x, (f x) ∂ (μs i)) (∫ x, (f x) ∂μ) | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hinf
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have key := h _ (f.add_norm_nonneg) | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hinf
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp_rw [f.integral_add_const ‖f‖] at key | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hinf
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul] at key | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hinf
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have := liminf_add_const L (fun i ↦ ∫ x, (f x) ∂ (μs i)) ‖f‖ bdd_above bdd_below | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hinf
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | rwa [this, add_le_add_iff_right] at key | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hsup
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have key := h _ (f.norm_sub_nonneg) | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hsup
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp_rw [f.integral_const_sub ‖f‖] at key | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hsup
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul] at key | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hsup
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have := liminf_const_sub L (fun i ↦ ∫ x, (f x) ∂ (μs i)) ‖f‖ bdd_above bdd_below | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.hsup
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | rwa [this, sub_le_sub_iff_left] at key | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x :... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | exact bdd_above | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
case inr.h'
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure Ω
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | exact bdd_below | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib.MeasureTheory.Measure.Portmanteau.546_0.COExymZuBXoOgED | lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ f : Ω →ᵇ ℝ, 0 ≤ f → ∫ x, (f x) ∂μ ≤ L.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)))
(f : Ω →ᵇ ℝ) :
Tendsto (fun i ↦ ∫ x,... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
h_opens :
∀ (G : Set Ω),
IsOpen G →
(fun s => ENNReal.toNNReal (↑↑↑μ s)) G ≤ liminf (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) G) atTop
⊢ Tendsto... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | refine ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mpr ?_ | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
h_opens :
∀ (G : Set Ω),
IsOpen G →
(fun s => ENNReal.toNNReal (↑↑↑μ s)) G ≤ liminf (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) G) atTop
⊢ ∀ (f : ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply tendsto_integral_of_forall_integral_le_liminf_integral | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
h_opens :
∀ (G : Set Ω),
IsOpen G →
(fun s => ENNReal.toNNReal (↑↑↑μ s)) G ≤ liminf (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) G) atTop
⊢ ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | intro f f_nn | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
h_opens :
∀ (G : Set Ω),
IsOpen G →
(fun s => ENNReal.toNNReal (↑↑↑μ s)) G ≤ liminf (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) G) atTop
f ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure (f := f) f_nn | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
h_opens :
∀ (G : Set Ω),
IsOpen G →
(fun s => ENNReal.toNNReal (↑↑↑μ s)) G ≤ liminf (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) G) atTop
f ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | intro G G_open | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
h_opens :
∀ (G : Set Ω),
IsOpen G →
(fun s => ENNReal.toNNReal (↑↑↑μ s)) G ≤ liminf (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) G) atTop
f ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | specialize h_opens G G_open | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : (fun s => ENNReal.toNNReal (↑↑↑μ s)) G ≤ liminf (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only at h_opens | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
⊢ ↑↑↑μ G ≤ lim... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have aux : ENNReal.ofNNReal (liminf (fun i ↦ ENNReal.toNNReal ((μs i : Measure Ω) G)) atTop) =
liminf (ENNReal.ofNNReal ∘ fun i ↦ (ENNReal.toNNReal ((μs i : Measure Ω) G))) atTop := by
refine Monotone.map_liminf_of_continuousAt (F := atTop) ENNReal.coe_mono (μs · G) ?_ ?_ ?_
· apply ENNReal.continuous... | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
⊢ ↑(liminf (fun i => ... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | refine Monotone.map_liminf_of_continuousAt (F := atTop) ENNReal.coe_mono (μs · G) ?_ ?_ ?_ | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case refine_1
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
⊢ Conti... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply ENNReal.continuous_coe.continuousAt | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case refine_2
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
⊢ IsBou... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | use 1 | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
⊢ ∀ᶠ (x : ℝ≥0)... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [eventually_map, ProbabilityMeasure.apply_le_one, eventually_atTop, ge_iff_le,
implies_true, forall_const, exists_const] | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case refine_3
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
⊢ IsBou... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | use 0 | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
⊢ ∀ᶠ (x : ℝ≥0)... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [zero_le, eventually_map, eventually_atTop, ge_iff_le, implies_true, forall_const,
exists_const] | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
aux :
↑(limi... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have obs := ENNReal.coe_mono h_opens | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
aux :
↑(limi... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [ne_eq, ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure, aux] at obs | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
aux :
↑(limi... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | convert obs | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
case h.e'_4.h.e'_4.h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
G : Set Ω
G_open : IsOpen G
h_opens : ENNReal.toNNReal (↑↑↑μ G) ≤ liminf (fun i => ENNReal.toNNReal (↑↑↑(μs i) G)) atTop
... | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | simp only [Function.comp_apply, ne_eq, ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure] | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib.MeasureTheory.Measure.Portmanteau.572_0.COExymZuBXoOgED | /-- One implication of the portmanteau theorem:
If for all open sets G we have the liminf condition `μ(G) ≤ liminf μsₙ(G)`, then the measures
μsₙ converge weakly to the measure μ. -/
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
{μs : ℕ → ProbabilityMeasure Ω}
(h_opens : ∀ G, IsOpen G → ... | Mathlib_MeasureTheory_Measure_Portmanteau |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
⊢ PartialOrder (Subobject X) | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | dsimp only [Subobject] | instance (X : C) : PartialOrder (Subobject X) := by
| Mathlib.CategoryTheory.Subobject.Basic.102_0.YX2OHoKUUy0oqDw | instance (X : C) : PartialOrder (Subobject X) | Mathlib_CategoryTheory_Subobject_Basic |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
⊢ PartialOrder (ThinSkeleton (MonoOver X)) | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | infer_instance | instance (X : C) : PartialOrder (Subobject X) := by
dsimp only [Subobject]
| Mathlib.CategoryTheory.Subobject.Basic.102_0.YX2OHoKUUy0oqDw | instance (X : C) : PartialOrder (Subobject X) | Mathlib_CategoryTheory_Subobject_Basic |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
p : Subobject X → Prop
h : ∀ ⦃A : C⦄ (f : A ⟶ X) [inst : Mono f], p (mk f)
P : Subobject X
⊢ p P | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | apply Quotient.inductionOn' | protected theorem ind {X : C} (p : Subobject X → Prop)
(h : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], p (Subobject.mk f)) (P : Subobject X) : p P := by
| Mathlib.CategoryTheory.Subobject.Basic.119_0.YX2OHoKUUy0oqDw | protected theorem ind {X : C} (p : Subobject X → Prop)
(h : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], p (Subobject.mk f)) (P : Subobject X) : p P | Mathlib_CategoryTheory_Subobject_Basic |
case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
p : Subobject X → Prop
h : ∀ ⦃A : C⦄ (f : A ⟶ X) [inst : Mono f], p (mk f)
P : Subobject X
⊢ ∀ (a : MonoOver X), p (Quotient.mk'' a) | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | intro a | protected theorem ind {X : C} (p : Subobject X → Prop)
(h : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], p (Subobject.mk f)) (P : Subobject X) : p P := by
apply Quotient.inductionOn'
| Mathlib.CategoryTheory.Subobject.Basic.119_0.YX2OHoKUUy0oqDw | protected theorem ind {X : C} (p : Subobject X → Prop)
(h : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], p (Subobject.mk f)) (P : Subobject X) : p P | Mathlib_CategoryTheory_Subobject_Basic |
case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
p : Subobject X → Prop
h : ∀ ⦃A : C⦄ (f : A ⟶ X) [inst : Mono f], p (mk f)
P : Subobject X
a : MonoOver X
⊢ p (Quotient.mk'' a) | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | exact h a.arrow | protected theorem ind {X : C} (p : Subobject X → Prop)
(h : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], p (Subobject.mk f)) (P : Subobject X) : p P := by
apply Quotient.inductionOn'
intro a
| Mathlib.CategoryTheory.Subobject.Basic.119_0.YX2OHoKUUy0oqDw | protected theorem ind {X : C} (p : Subobject X → Prop)
(h : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], p (Subobject.mk f)) (P : Subobject X) : p P | Mathlib_CategoryTheory_Subobject_Basic |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
p : Subobject X → Subobject X → Prop
h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [inst : Mono f] [inst_1 : Mono g], p (mk f) (mk g)
P Q : Subobject X
⊢ p P Q | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | apply Quotient.inductionOn₂' | protected theorem ind₂ {X : C} (p : Subobject X → Subobject X → Prop)
(h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g],
p (Subobject.mk f) (Subobject.mk g))
(P Q : Subobject X) : p P Q := by
| Mathlib.CategoryTheory.Subobject.Basic.126_0.YX2OHoKUUy0oqDw | protected theorem ind₂ {X : C} (p : Subobject X → Subobject X → Prop)
(h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g],
p (Subobject.mk f) (Subobject.mk g))
(P Q : Subobject X) : p P Q | Mathlib_CategoryTheory_Subobject_Basic |
case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
p : Subobject X → Subobject X → Prop
h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [inst : Mono f] [inst_1 : Mono g], p (mk f) (mk g)
P Q : Subobject X
⊢ ∀ (a₁ a₂ : MonoOver X), p (Quotient.mk'' a₁) (Quotient.mk'' a₂) | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | intro a b | protected theorem ind₂ {X : C} (p : Subobject X → Subobject X → Prop)
(h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g],
p (Subobject.mk f) (Subobject.mk g))
(P Q : Subobject X) : p P Q := by
apply Quotient.inductionOn₂'
| Mathlib.CategoryTheory.Subobject.Basic.126_0.YX2OHoKUUy0oqDw | protected theorem ind₂ {X : C} (p : Subobject X → Subobject X → Prop)
(h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g],
p (Subobject.mk f) (Subobject.mk g))
(P Q : Subobject X) : p P Q | Mathlib_CategoryTheory_Subobject_Basic |
case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
p : Subobject X → Subobject X → Prop
h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [inst : Mono f] [inst_1 : Mono g], p (mk f) (mk g)
P Q : Subobject X
a b : MonoOver X
⊢ p (Quotient.mk'' a) (Quotient.mk'' b) | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | exact h a.arrow b.arrow | protected theorem ind₂ {X : C} (p : Subobject X → Subobject X → Prop)
(h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g],
p (Subobject.mk f) (Subobject.mk g))
(P Q : Subobject X) : p P Q := by
apply Quotient.inductionOn₂'
intro a b
| Mathlib.CategoryTheory.Subobject.Basic.126_0.YX2OHoKUUy0oqDw | protected theorem ind₂ {X : C} (p : Subobject X → Subobject X → Prop)
(h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g],
p (Subobject.mk f) (Subobject.mk g))
(P Q : Subobject X) : p P Q | Mathlib_CategoryTheory_Subobject_Basic |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
A : C
X Y : Subobject A
h : X = Y
⊢ eqToHom (_ : underlying.obj X = underlying.obj Y) ≫ arrow Y = arrow X | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | induction h | @[simp]
theorem arrow_congr {A : C} (X Y : Subobject A) (h : X = Y) :
eqToHom (congr_arg (fun X : Subobject A => (X : C)) h) ≫ Y.arrow = X.arrow := by
| Mathlib.CategoryTheory.Subobject.Basic.210_0.YX2OHoKUUy0oqDw | @[simp]
theorem arrow_congr {A : C} (X Y : Subobject A) (h : X = Y) :
eqToHom (congr_arg (fun X : Subobject A => (X : C)) h) ≫ Y.arrow = X.arrow | Mathlib_CategoryTheory_Subobject_Basic |
case refl
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
A : C
X Y : Subobject A
⊢ eqToHom (_ : underlying.obj X = underlying.obj X) ≫ arrow X = arrow X | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | simp | @[simp]
theorem arrow_congr {A : C} (X Y : Subobject A) (h : X = Y) :
eqToHom (congr_arg (fun X : Subobject A => (X : C)) h) ≫ Y.arrow = X.arrow := by
induction h
| Mathlib.CategoryTheory.Subobject.Basic.210_0.YX2OHoKUUy0oqDw | @[simp]
theorem arrow_congr {A : C} (X Y : Subobject A) (h : X = Y) :
eqToHom (congr_arg (fun X : Subobject A => (X : C)) h) ≫ Y.arrow = X.arrow | Mathlib_CategoryTheory_Subobject_Basic |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
P : Subobject X
Q : MonoOver X
⊢ mk (arrow (Quotient.mk'' Q)) = Quotient.mk'' Q | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | obtain ⟨e⟩ := @Quotient.mk_out' _ (isIsomorphicSetoid _) Q | @[simp]
theorem mk_arrow (P : Subobject X) : mk P.arrow = P :=
Quotient.inductionOn' P fun Q => by
| Mathlib.CategoryTheory.Subobject.Basic.258_0.YX2OHoKUUy0oqDw | @[simp]
theorem mk_arrow (P : Subobject X) : mk P.arrow = P | Mathlib_CategoryTheory_Subobject_Basic |
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