chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
13,131 | lean_file_chunk | 9f154c1f33cc5324e2fd0e336eb0845159c2af995e6d300d4c7d4316f3c3d72b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Measure.AbsolutelyContinuous
/-!
# Vitali families
On a metric space `X` with a measure `μ`, consider for each `x ... |
8,183 | lean_file_chunk | ba1e6627aa7e685c331d173deef5f4c681cc2ae478d9b6afce293ca4a7bb4275 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
module
public import Mathlib.MeasureTheory.Measure.Doubling
public import Mathlib.MeasureTheory.Covering.Vitali
public import Mathlib.MeasureTheory.Covering.Differentiatio... |
7,018 | lean_file_chunk | 3fcf543f40e0aedda04818c241751e44e5fd1774e7a923a364acea3a07d5394b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.Analysis.SpecialFunctions.PolarCoord
public import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
/-!
# Integrals involving the Gamma func... |
23,970 | lean_file_chunk | 69182c034217e2cee2bbeb7784ca067f34753cd72d3bebc202e29ff654bb8f80 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
/-!
# Additivity on measurable sets with finite... |
1,164 | lean_file_chunk | 4b613af67aed9a46e2b49fbd13a31934fd541c24279c055ce65436470272a6e7 | lean_source_file | pkg_mathlib_source_file_chunks | simp only [coe_piecewise, piecewise_eq_indicator, coe_const, Function.const_zero,
piecewise_eq_indicator]
rw [indicator_preimage, ← Function.const_def, preimage_const_of_mem]
swap; · exact Set.mem_singleton x
rw [← Function.const_zero, ← Function.const_def, preimage_const_of_notMem]
swap; · rw [Set.mem_sing... |
23,969 | lean_file_chunk | 6f3d02f53916b4d97f29f5ef091600366cd407d7e5ecf9eb19daac6983e8710b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Function.L1Space.Integrable
public import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions ... |
19,042 | lean_file_chunk | 20f5ecde4223f6f8810fcab2c15da169a1e4cc0bace3fd7f89f25d6d586e3ed3 | lean_source_file | pkg_mathlib_source_file_chunks | (hf : IntegrableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
IntegrableAtFilter (f + g) l μ := by
rcases hf with ⟨s, sl, hs⟩
rcases hg with ⟨t, tl, ht⟩
refine ⟨s ∩ t, inter_mem sl tl, ?_⟩
exact (hs.mono_set inter_subset_left).add (ht.mono_set inter_subset_right)
protected theorem IntegrableAtFilter.ne... |
1,876 | lean_file_chunk | a3b12c0fe4bc899ca2096b573b591175ab598e87c85c4839798b63adb77bb81a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
module
public import Mathlib.MeasureTheory.Integral.Asymptotics
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
public import Mathlib.MeasureTheo... |
8,598 | lean_file_chunk | 7961acaa7ba836476a6080ef91cbbbe7f3dafe04b3f8d1c55521702566cd8628 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.MeasureTheory.Integral.Bochner.Basic
public import Mathlib.Topology.ContinuousMap.Bounded.Normed
public import Mathlib.Topology.Algebra.Orde... |
14,584 | lean_file_chunk | ec1302f2610afb292ff8871637b739ec75e3bea7bfdbc3f1d69a5f73673842df | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Stefan Kebekus. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stefan Kebekus
-/
module
public import Mathlib.MeasureTheory.Integral.CircleIntegral
public import Mathlib.MeasureTheory.Integral.IntervalAverage
public import Mathlib.MeasureTheory.I... |
3,654 | lean_file_chunk | 77d1f7142e4884e2e3c62dc85835fbc0cc01e13fa51eff00c42f2b59c30b33cd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Integral.Bochner.Set
public import Mathlib.Topology.UrysohnsLemma
/-!
# Integrals of continuous functions with resp... |
4,623 | lean_file_chunk | e252b887623d567413d87813af15c28759263cf300a94c303e7189d13537d07c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
public import Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence
/-!
# Results ... |
9,418 | lean_file_chunk | bcec8a8795a39586482e2ec7e706f9c2dacacf9aab07e34bdd55cccae9d9351e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Lawrence Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lawrence Wu
-/
module
public import Mathlib.MeasureTheory.Group.Measure
public import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Bounding of integrals by asymptoti... |
1,896 | lean_file_chunk | 4b247c8154f920f1d67c361ec3bca32144a96477c760d4cb64a08aff0dead38c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Measure.WithDensity
public import Mathlib.Analysis.Normed.Module.Basic
/-! # A lemma about measurability with densi... |
11,325 | lean_file_chunk | 056ff2fdcdaa74da3f97ec9f0656fd7f76a583018245287c98475294c6292928 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Cuma Kökmen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Cuma Kökmen, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Integral.CircleIntegral
public import Mathlib.MeasureTheory.Integral.Prod
public import Mathlib.Order.Fin.Tuple... |
6,447 | lean_file_chunk | f586dcd4f7f110f16e15a79dcff6795fe30ce19b375b5c7d7b94d69bf55f17d6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
module
public import Mathlib.Data.Complex.Basic
public import Mathlib.MeasureTheory.Integral.CircleIntegral
/-!
# Circle integral transform
In this file we define ... |
23,939 | lean_file_chunk | 913dc71f1976ad1546dc0b94f310ee0bca45ce74d874fe0208c2762a3872cd62 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies, Louis (Yiyang) Liu
-/
module
public import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this fi... |
15,564 | lean_file_chunk | 0ba5d5af48696a10b2b674e8122b9174f67a4226ce77c50349508898e758f46a | lean_source_file | pkg_mathlib_source_file_chunks | /-- **First moment method**. The maximum of an integrable function is greater than its mean. -/
theorem exists_setAverage_le (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) :
∃ x ∈ s, ⨍ a in s, f a ∂μ ≤ f x :=
let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_setAverage_le_pos hμ hμ₁ hf).ne'
⟨x, h... |
23,950 | lean_file_chunk | 3503e016784f416dbb995fb5f3309e26efe37938e2f7c6d800111fec5530aa65 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Patrick Massot, Louis (Yiyang) Liu
-/
module
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
public import Mathlib.MeasureTheory.Mea... |
13,730 | lean_file_chunk | 9a14d8a545b7963d87a57a185351efcc608ca6180823b7ef72b8d83489bc779e | lean_source_file | pkg_mathlib_source_file_chunks | · have : ∀ x ∈ Icc a b, ∫ t in Ioc a x, f t ∂μ = ∫ t in a..x, f t ∂μ := by
intro x x_in
simp_rw [integral_of_le x_in.1]
rw [continuousOn_congr this]
intro x₀ _
refine continuousWithinAt_primitive (measure_singleton x₀) ?_
simp only [intervalIntegrable_iff_integrableOn_Ioc_of_le, max_eq_right... |
23,960 | lean_file_chunk | 3cad836b64987c71946816bbf70c793915492652b9ec85b0024d459d8704e279 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.MeasureTheory.Function.LpSeminorm.Prod
public import Mathlib.MeasureTheory.Integral.DominatedConvergence
public import Mathlib.Measu... |
7,691 | lean_file_chunk | ab90d6186faf86c91eadbfc48f29687c07982a9566cec66e092e93d325b9cc1a | lean_source_file | pkg_mathlib_source_file_chunks | /-- Symmetric version of **Fubini's Theorem**: For integrable functions on `α × β`,
the Bochner integral of `f` is equal to the iterated Bochner integral.
This version has the integrals on the right-hand side in the other order. -/
theorem integral_prod_symm (f : α × β → E) (hf : Integrable f (μ.prod ν)) :
∫ z,... |
7,378 | lean_file_chunk | ac09d8a9554b9537be63f48ba7a962701096fe8235fe2fe1d20ac58173dc9523 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.MeasureTheory.Integral.Prod
/-!
# Integration with respect to a finite product of measures
On a finite product of measure spaces, we sho... |
1,827 | lean_file_chunk | 15e53d78b3a62bc309a80e5096d75883a8fd5208cf0cae638ddc79836315ab40 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yongxi Lin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yongxi Lin, Thomas Zhu
-/
module
public import Mathlib.MeasureTheory.Function.LocallyIntegrable
public import Mathlib.MeasureTheory.Integral.Bochner.Basic
public import Mathlib.Topology.C... |
4,628 | lean_file_chunk | e03941918dbdc5f3c1407516443ad08e31480391d07d725d18f7beabc25b1685 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Louis (Yiyang) Liu
-/
module
public import Mathlib.MeasureTheory.Integral.Average
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
/-!
# Integral... |
21,558 | lean_file_chunk | a8236bb5453701e1337b9ef6b086be1271e68380cdfdcbd813af21ad13996bf1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.Analysis.MeanInequalities
public import Mathlib.Analysis.MeanInequalitiesPow
public import Mathlib.MeasureTheory.Function.SpecialFunctions.B... |
23,958 | lean_file_chunk | 684ab8c4b5df3ceb38ba6b86b748f54d23ce89f8afe820abf681c2f37bd80ada | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
module
public import Mathlib.Analysis.Calculus.Deriv.Support
public import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
public import Mathlib.Measur... |
23,996 | lean_file_chunk | 5541f6d7a303f0f6977dd4fbc088b6c0b0209b4f7cdc450898947be40b204c1e | lean_source_file | pkg_mathlib_source_file_chunks | theorem AECover.integral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α}
(hφ : AECover μ l φ) {f : α → E} (I : E) (hfi : Integrable f μ)
(h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I :=
tendsto_nhds_unique (hφ.integral_tendsto_of_countably_generated hfi) h
theorem AE... |
19,579 | lean_file_chunk | 5382464c51744746e5e31d6a9e9be9013de9e8cb768fbc40a4a43c8719436476 | lean_source_file | pkg_mathlib_source_file_chunks | theorem integral_Iic_of_hasDerivAt_of_tendsto'
(hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
(hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m := by
refine integral_Iic_of_hasDerivAt_of_tendsto ?_ (fun x hx => hderiv x hx.out.le)
f'int hf
exact (hderiv a self... |
11,659 | lean_file_chunk | 802e8a44d4c743dd4e90549ba480a2dcf53d13db8dcff7234931cce5b57b0f49 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
module
public import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way... |
23,916 | lean_file_chunk | 35b0c0c8d3de20debf83e021cdd0603ae02fc634b3ce421bc4c7c64aba320e94 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.Analytic.IsolatedZeros
public import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
public import Mathlib.Analysis.Speci... |
9,103 | lean_file_chunk | 0c56e36ab933d15c75be72df3a25678f300757d1133ff15a0e517a681ca745f7 | lean_source_file | pkg_mathlib_source_file_chunks | (h : ∀ z ∈ sphere c R, HasDerivWithinAt f (f' z) (sphere c R) z) : (∮ z in C(c, R), f' z) = 0 :=
integral_eq_zero_of_hasDerivWithinAt' <| (abs_of_nonneg hR).symm ▸ h
/-- If `n < 0` and `|w - c| = |R|`, then `(z - w) ^ n` is not circle integrable on the circle with
center `c` and radius `|R|`, so the integral `∮ z in... |
23,954 | lean_file_chunk | 1b9402e73f15a71768d2b74a26e49e16ce1aa77846b5b9358f4d18bf6ce55c9f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
/-!
# The layer cake formula / Cavalieri's principle / tail probability formula
In this file... |
4,409 | lean_file_chunk | 74259f394c9fdbef06d4e57a6880aceaa14204e8d08d4ff392be3088b45439fe | lean_source_file | pkg_mathlib_source_file_chunks | Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in 0..∞, g(t) * μ {ω | f(ω) > t}`.
See `lintegral_comp_eq_lintegral_meas_le_mul` for a version with sets of the form `{ω | f(ω) ≥ t}`
instead. -/
theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
(f_mble : AEMeasurable f μ) ... |
23,971 | lean_file_chunk | c419d8f74f3c2fb10c3e6d51552947bd35f9825fc584caba2c293b4bbfa5f917 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Integral.FinMeasAdditive
public import Mathlib.Analysis.Normed.Operator.Ext... |
23,963 | lean_file_chunk | 673363401251f5721532a3210060fa72eaa196fd535b1e7bf569832249714c0a | lean_source_file | pkg_mathlib_source_file_chunks | exact setToL1S_indicatorConst (fun s => hT.eq_zero_of_measure_zero) hT.1 hs hμs x
theorem setToL1_indicatorConstLp (hT : DominatedFinMeasAdditive μ T C) {s : Set α}
(hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : E) :
setToL1 hT (indicatorConstLp 1 hs hμs x) = T s x := by
rw [← Lp.simpleFunc.coe_indicatorConst h... |
7,343 | lean_file_chunk | c1ae0d55e1aff6792b6fbebcd12c7c64ac9d2f0e196d65c771aed26247faa5ea | lean_source_file | pkg_mathlib_source_file_chunks | (f : α → E) : setToFun μ T hT f = setToFun (c • μ) T hT_smul f := by
by_cases hc0 : c = 0
· simp [hc0] at hT_smul
have h : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0 := fun s hs _ => hT_smul.eq_zero hs
rw [setToFun_zero_left' _ h, setToFun_measure_zero]
simp [hc0]
refine setToFun_congr_measure c⁻¹ c ?_ ... |
5,188 | lean_file_chunk | ace061547e48c8497c578fb383d5ca62431668603243a73bdbc80d1ee0b63ada | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Louis (Yiyang) Liu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis (Yiyang) Liu
-/
module
public import Mathlib.MeasureTheory.Integral.Average
/-!
# First mean value theorem for set integrals
We prove versions of the first mean value theo... |
23,920 | lean_file_chunk | 2d3534c85652981ce5510cbe414010a0200192e39ea06685015f23f339505507 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Integral.IntegralEqImproper
/-!
# Integrals against peak functions
A sequence of peak functions is a sequence of f... |
1,808 | lean_file_chunk | 20df0d26147efb90cc0cafdf90cdf34b790d4eeafabdfa9e2c90548a2aed1d24 | lean_source_file | pkg_mathlib_source_file_chunks | exact (aecover_closedBall tendsto_id).integral_tendsto_of_countably_generated I
apply this.congr'
filter_upwards [Ioi_mem_atTop 0] with c (hc : 0 < c)
rw [integral_const_mul, setIntegral_comp_smul_of_pos _ _ _ hc, smul_eq_mul, ← mul_assoc,
mul_inv_cancel₀ (by positivity), _root_.smul_closedBall _ _ ze... |
23,949 | lean_file_chunk | 79a52f12d0fb4beac6f05186e6298b5ba05dc83d3ece67208f213d56dbfc0aa2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.BoxIntegral.DivergenceTheorem
public import Mathlib.Analysis.BoxIntegral.Integrability
public import Mathlib.Analysis.Calculu... |
7,990 | lean_file_chunk | ed501f2e003c184061404095c46c79350a68d562f9cfc09638fd1327d6914b84 | lean_source_file | pkg_mathlib_source_file_chunks | let e : (ℝ × ℝ) ≃L[ℝ] ℝ² := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
calc
(∫ x in Icc a b, f' x (1, 0) + g' x (0, 1)) =
∑ i : Fin 2,
((∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
![f, g] i (e.symm <| i.insertNth (e b i) x)) -
∫ x in Icc (e a ∘ i.succAbove) (... |
1,719 | lean_file_chunk | 65ece768e8855104d733b7ad966fd3ac8256275a89f31f2696571aeeb73d4e59 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.Algebra.Order.Group.PosPart
public import Mathlib.MeasureTheory.Group.Arithmetic
public import Mathlib.MeasureTheory.Order.Lattice
/-!
# ... |
3,812 | lean_file_chunk | 7e17da671e1d7a09135a9106d957f2be745571e7cb3872265157c5537f0eb6d2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Oliver Butterley. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Butterley, Yoh Tanimoto
-/
module
public import Mathlib.MeasureTheory.VectorMeasure.Variation.Defs
/-!
# Properties of variation
We prove basic properties of `variation` fo... |
2,110 | lean_file_chunk | 77d8b11442d6bc2f38cb51ecf3b08e49addf833427bd921b76049012b6791c9c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Oliver Butterley. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Butterley, Yoh Tanimoto
-/
module
public import Mathlib.Analysis.Normed.Group.InfiniteSum
public import Mathlib.MeasureTheory.Measure.PreVariation
/-!
# Total variation for ... |
23,662 | lean_file_chunk | 62da6288d0ffe0a91adc6dcf39c68f3e511b91cede92b726d1c78276a60363c3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
module
public import Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn
public import Mathlib.MeasureTheory.Measure.MutuallySingular
public import Mathlib.Topology.Alg... |
5,292 | lean_file_chunk | db3926ec5d675df9863bdb44c6fb542a239f22ce68f4f1f7b3c4b3a75064e31a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Loic Simon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Loic Simon
-/
module
public import Mathlib.MeasureTheory.Measure.Decomposition.Hahn
public import Mathlib.MeasureTheory.Measure.Sub
public import Mathlib.MeasureTheory.VectorMeasure.Decom... |
20,103 | lean_file_chunk | 42d58f940ea0a96c1eaa0502d5086be34163d84af48f0f05b9d40f4a004ecd9c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
module
public import Mathlib.MeasureTheory.VectorMeasure.Basic
public import Mathlib.Order.SymmDiff
/-!
# Hahn decomposition
This file proves the Hahn decomposition theo... |
2,536 | lean_file_chunk | 88016ec8cdc3a21036f036c7367e153f5894098567dcf0a8982447a9b9744a8c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Thomas Zhu
-/
module
public import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
public import Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue
/-!
# R... |
23,486 | lean_file_chunk | d9e0c9e6e2c0770ebd72128d3e4d822eeaf3c8cd2c3397ba3fbf5e72a91e7905 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
module
public import Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue
public import Mathlib.MeasureTheory.Measure.Complex
public import Mathlib.MeasureTheory.VectorMea... |
3,309 | lean_file_chunk | 410358ee3f2ecda4538cf9f1309ae90833be9f31597ddd1774d6d555d3cd7176 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.Analysis.Real.Cardinality
public import Mathlib.MeasureTheory.Constructions.Polish.Basic
/-!
# A Polish Borel space is measurably equivalen... |
3,953 | lean_file_chunk | 67ca3594f3da3afbb98e88f7ee67862857d9abbcf88335737b78d1d66969e8ae | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Etienne Marion
-/
module
public import Mathlib.MeasureTheory.Constructions.Polish.Basic
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
/-!
# Results about stro... |
23,972 | lean_file_chunk | 8269139acdda25680053bd9b5fe695e05618130d1a09927c1fb2365f63f32cd3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Felix Weilacher
-/
module
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
... |
23,989 | lean_file_chunk | bf770a5d6ee80bee88f90aaed362d27a76a2b0756cc0b0c712e3496b6a267bf9 | lean_source_file | pkg_mathlib_source_file_chunks | complement is measurable. -/
theorem AnalyticSet.measurableSet_of_compl [T2Space α] [MeasurableSpace α] [OpensMeasurableSpace α]
{s : Set α} (hs : AnalyticSet s) (hsc : AnalyticSet sᶜ) : MeasurableSet s := by
rcases hs.measurablySeparable hsc disjoint_compl_right with ⟨u, hsu, hdu, hmu⟩
obtain rfl : s = u := hs... |
4,560 | lean_file_chunk | 0ed6da8239c534fb21640f4fb3033de9ae351c2a44cd6440dc250f98c2e66b5c | lean_source_file | pkg_mathlib_source_file_chunks | {f : ι → β → γ} (hf : ∀ i, Measurable (f i)) {g : β → γ} (hg : Measurable g) :
MeasurableSet { x | Tendsto (fun n ↦ f n x) l (𝓝 (g x)) } := by
letI := TopologicalSpace.pseudoMetrizableSpacePseudoMetric γ
simp_rw [tendsto_iff_dist_tendsto_zero (f := fun n ↦ f n _)]
exact measurableSet_tendsto (𝓝 0) (fun n ↦ ... |
2,976 | lean_file_chunk | 573a4ce82b32a9538e9c64ec5cc022927fa5a8b70be992a59dc2945b828f065b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
public import Mathlib.Topology.Order.WithTop
/-!
# Borel measurable space on `WithTop`
For `ι... |
3,277 | lean_file_chunk | c8e4ca1bd2835620676a02f871a2a961240f001cf03d73786f97e8c8d8bae20f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
module
public import Mathlib.Analysis.Normed.Operator.Bilinear
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
public import Mathlib.Topology.Alge... |
10,270 | lean_file_chunk | eab6590b817d76a31814691393d2afc9263dc8fc5b2ad7e1d2d000f15aaf7387 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
module
public import Mathlib.Analysis.Normed.Group.Continuity
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
public import Mathl... |
7,544 | lean_file_chunk | 33333a90d730e1e9351858245ce6b2acbce86acc3b769360591b0d8d0d7f1955 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
public import Mathli... |
23,970 | lean_file_chunk | b2b903ae0e649aae18e71fcdd92b5ac6819f4174d0d0dfeb454d9ff333492c9e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
module
public import Mathlib.Topology.Semicontinuity.Basic
public import Mathlib.MeasureTheory.Function.AEMeasurableSequence
public imp... |
23,976 | lean_file_chunk | 48734eea4a7cf90573a9c3784eaf87a1ed0a587c7417ccdf1e8a12b3d6f759df | lean_source_file | pkg_mathlib_source_file_chunks | · rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, -, hst⟩
have : DirectedOn (· ≤ ·) s := directedOn_iff_directed.2 (Subtype.mono_coe _).directed_le
simp only [← biSup_measure_Iic hsc (hsd.exists_ge' hst) this, h]
rw [← Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) nullMeasurableSet_Iic,
... |
2,230 | lean_file_chunk | 847e3dbc3dc371b5e00534eacaebd38f0d3e134d1bcec18d56c65d1ca2a2dd10 | lean_source_file | pkg_mathlib_source_file_chunks | gives a way to compute the measure of a set in terms of sets on which a given function `f` does not
fluctuate by more than `t`. -/
theorem measure_eq_measure_preimage_add_measure_tsum_Ico_zpow {α : Type*} {mα : MeasurableSpace α}
(μ : Measure α) {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s)
... |
23,961 | lean_file_chunk | 9305569b6bab829a54122ae5d8db4ca27796064471452638492a7f446dd8ae16 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Group.Arithmetic
public import Mathlib.Topology.GDelta.MetrizableSpace
public import Mathlib.Topology.Ins... |
8,422 | lean_file_chunk | 35ef4ab0160ab5dcada76e2fa930fbd6cf516f34f7949b9c6471da348f21c67d | lean_source_file | pkg_mathlib_source_file_chunks | MeasurableInv γ := ⟨continuous_inv.measurable⟩
@[to_additive]
instance (priority := 100) ContinuousConstSMul.toMeasurableConstSMul {M α} [TopologicalSpace α]
[MeasurableSpace α] [BorelSpace α] [SMul M α] [ContinuousConstSMul M α] :
MeasurableConstSMul M α where
@[to_additive]
instance (priority := 100) Contin... |
698 | lean_file_chunk | 044eced62c89fc6f6cf938abe9f95937a41cdc349e3a985ff31c87377fa130ab | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.Complex.Basic
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
/-! # Equip `ℂ` with the Borel sigma-algebr... |
23,996 | lean_file_chunk | a1901f1a4aa20cb8098e66a95a5e91e204bf2c15f93a6b58c011d8aeb96d5ee4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
public import Mathlib.MeasureTheory.MeasurableSpace.Prod
public import Mat... |
1,531 | lean_file_chunk | 90d59c47d15fa7aee8b0c4bcf469acb7e15b5039b39c4c64d5c27861e8fe154f | lean_source_file | pkg_mathlib_source_file_chunks | ⋃ i, sigma_finite_sets i ∩ norm_sets i = (⋃ i, sigma_finite_sets i) ∩ ⋃ i, norm_sets i := by
refine Set.iUnion_inter_of_monotone (monotone_spanningSets μ) fun i j hij x => ?_
simp only [norm_sets, Set.mem_setOf_eq]
refine fun hif => hif.trans ?_
exact mod_cast hij
rw [this, norm_sets_spannin... |
12,995 | lean_file_chunk | 86d981f4653e21eea19872fdd1100d9ec7313f8d06abdff1edbf8698848d74fe | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Yongxi Lin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yongxi Lin, Thomas Zhu
-/
module
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.Analysis.Convex.Approximation
import Mathlib.Analysis.Convex... |
23,517 | lean_file_chunk | 6d0407dd6a6b0bd36d5800ea1f9b521de6733fd9932269ee1f76586c4c4ad952 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
public import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L... |
9,185 | lean_file_chunk | fd7f99eb8a6524e06cfc25edcd57cb4bfa215167d61556560f81fb87df7bc7bc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
/-!
# Conditional expectation of indicator functions
This file proves some results ab... |
13,931 | lean_file_chunk | 8ef0c75d76483efcfded412f766ea5f06790acea93f915e5502a4b8bc87aded4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Etienne Marion, Rémy Degenne, Kexing Ying
-/
module
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
import Mathlib.MeasureTheory.Function.ConditionalExpectat... |
23,472 | lean_file_chunk | 71d0d96715698da7e6335d50d787862038531615941f10e1c045129c7e061622 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrabl... |
8,132 | lean_file_chunk | f34abd75d164dd06ac826155a7981eb1f4f5786ffdfe06b95aa12d9485388370 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
public import Mathlib.MeasureTheory.Function.ConditionalLExpectation
public import Mathl... |
2,190 | lean_file_chunk | c50d61ce426224724f564c77d38b324a222e0223b45734886180ccfcc017eb02 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
public import Mathlib.MeasureTheory.Function.ConditionalLExpectation
/-!
# Results abo... |
9,536 | lean_file_chunk | ae62b0eadd2842f4e18b6988fbddf7276c0a8a6761b5f17d2547d5f5cb0e066c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.AEEqOfIntegral
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable
/-!
# Uniqueness of ... |
23,775 | lean_file_chunk | f24320f5f22e0db5a6882c34f86be04efe06f00508a8df881d20d5c75cf64041 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
/-! # Functions a.e. measurable w... |
22,903 | lean_file_chunk | 014fd6729d0aa40b7a3dd77c826c4246e3e27d8bf9cbdfcdb05b29993b032c38 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.Analysis.InnerProductSpace.Projection.Basic
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
public import Mathlib... |
10,707 | lean_file_chunk | 86481e7ccbd6a7777d569718ad8cf08059ba58ee80c561ad432c31528b506a61 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
module
public import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
public import Mathlib.MeasureTheory.Function.UniformIntegrable
public i... |
2,638 | lean_file_chunk | aa1e6abbe8866f93c3216a3b9da060ea0a01c9aa5cdb9cd8bbab72b5942f2986 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas
/-!
# Finitely strongly mea... |
4,931 | lean_file_chunk | 67cdf6cd36a7cf268f6881e5cf5b7250bfd0a5b81bfd5d3c90e17c49e395d972 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
public import Mathlib.Dynamics.Ergodic.MeasurePreserving
public import Mathlib... |
1,196 | lean_file_chunk | d64fc8e2edac5ac49e31b07ad69f450890a4463b15165c9438f4315dfe6ae12a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
public import Mathlib.MeasureTheory.Integral.Lebesgue.Add
/-!
# Finitely str... |
2,041 | lean_file_chunk | c9b72d0cfc1f97f193d82fc3d5a033b0970c1ab34dc63f1f51abe1ae2d7ca8a0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.InnerProductSpace.Continuous
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurabl... |
23,957 | lean_file_chunk | 8a655776b6a757664af8693df8c10f5c2d4d4e54ee357adbf16fe2b5e2a9c291 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Module.Basic
public import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and ... |
23,924 | lean_file_chunk | 1a7c909d85bc10f48654f299ac3bce56f49cd7f34b83c4febbe42b1faaf3c2a5 | lean_source_file | pkg_mathlib_source_file_chunks | variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
theorem _root_.stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
⟨fun h => by simpa only [inv_smul... |
13,114 | lean_file_chunk | 9ea6f02b06e37756d0f4c48de12f003e034775048a56a43279e680bd98717609 | lean_source_file | pkg_mathlib_source_file_chunks | Set.inter_union_distrib_left, Set.inter_comm (g_seq_s n ⁻¹' {x})]
refine MeasurableSet.union (hs _ (hs_m.inter ?_)) ?_
· exact @SimpleFunc.measurableSet_fiber _ _ m _ _
by_cases hx : x = 0
· suffices g_seq_s n ⁻¹' {x} ∩ sᶜ = sᶜ by
rw [this]
exact hs_m₂.compl
... |
23,984 | lean_file_chunk | 4e88d00395fb45fd96b1c28059e5e6692f628b184dc648282a8af4cecbce03e7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
/-!
# Strongly measurable and finitely strongly measurable functions
A ... |
18,237 | lean_file_chunk | af227fed31692759c70ee415e32964dc14c960524f6e43e0477ca7be2659a379 | lean_source_file | pkg_mathlib_source_file_chunks | change (hf.mk f x < hg.mk g x) = (f x < g x)
simp only [hfx, hgx]
theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { a | f a ≤ g a } μ := by
apply
(hf.stronglyMea... |
8,019 | lean_file_chunk | 11dd2d7fc65f18e4d647f8cd802046b22372ef3bc71cf831651a20f658a68fe0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
module
public import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# `L¹` space
In this file we establish an API between `Integrable` and the space `L¹` of e... |
22,939 | lean_file_chunk | 12ba065baa44bed829dbf3297b5729c82ade3b6540623a722d39f4983dd3c013 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
module
public import Mathlib.Analysis.Normed.Order.Lattice
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
public import Mathlib.M... |
23,991 | lean_file_chunk | 083f99a5bea4085f08cee3ed8412a46d227f4ee7015db0c0d1ca004d78d5b18e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
module
public import Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral
public import Mathlib.MeasureTheory.Function.LpOrder
public import Mathlib.MeasureTheory.... |
23,934 | lean_file_chunk | 759aa5bc4379e5a75bd7b4b5ad29c43782472e29b399c72c439bce087f773fd9 | lean_source_file | pkg_mathlib_source_file_chunks | exact hf.sup hg
@[fun_prop]
theorem Integrable.abs {β}
[NormedAddCommGroup β] [Lattice β] [HasSolidNorm β] [IsOrderedAddMonoid β]
{f : α → β} (hf : Integrable f μ) :
Integrable (fun a => |f a|) μ := by
rw [← memLp_one_iff_integrable] at hf ⊢
exact hf.abs
-- TODO: generalise the following lemmas to eno... |
5,340 | lean_file_chunk | 1f7e42e9a1c4d7abd4aad58c5edbdb4813af9d03451f0f9986dd99ecec220cf6 | lean_source_file | pkg_mathlib_source_file_chunks | theorem Integrable.im (hf : Integrable f μ) : Integrable (fun x => RCLike.im (f x)) μ := by
rw [← memLp_one_iff_integrable] at hf ⊢
exact hf.im
end RCLike
section Trim
variable {H : Type*} [NormedAddCommGroup H] {m0 : MeasurableSpace α} {μ' : Measure α} {f : α → H}
theorem Integrable.trim (hm : m ≤ m0) (hf_int ... |
2,216 | lean_file_chunk | 4d9309c6b173f0d8ce90832c8ae5cb0807ad446b649ebbd1991277241c4e163b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.RCLike.Lemmas
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
/-!
# Measurability of the basic `RCLike`... |
698 | lean_file_chunk | fac964012ec1197b293a4019236747cc477de1e8961bceacfac9b38a1d9ed71d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
/-!
# Measurability... |
2,235 | lean_file_chunk | 602926bf511814798cb55de17a5f29e654a9fad1dc4db18bb17628a61346747e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.InnerProductSpace.Continuous
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
/-!
# Measurability of sca... |
9,540 | lean_file_chunk | 67b21ccca6b7db67a4723e3c1fd19caee65c2852b2d3159f9205d0ec2dd97db1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
public import Mathlib.Measur... |
1,900 | lean_file_chunk | eee1daeaf8c1b467ebd76b0187fe0d3e60e0f478431ffea81423495115fd68f9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc
public import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
public import Mathlib.Measu... |
19,978 | lean_file_chunk | 3fbc0f04605fd634da12b2c1bb4a9c0c4ef785c94230104c7a2f1ef2ad449448 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
public import Mathlib.MeasureTheory.Function.LpSpace.Basic
/-!
# `Lp` is a comp... |
14,434 | lean_file_chunk | 6b67407208277612d6a9a98812b071545ab584fbdabab1956a1d95de64215793 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
public import Mathlib.MeasureTheory.Function.LpSpace.Basic
public import Mathlib... |
23,983 | lean_file_chunk | 93ff3c9c16fdd5263f4e5416efe28ff8ab3a51b54097fe38998748aa24611d6a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Operator.NNNorm
public import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
public import Mat... |
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