Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
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import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal NNReal Filter Set Fu... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 42 | 61 | theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, ∀ i, p.2 ∈ K i → closedBall p.2 p.1 ⊆ U i := by |
suffices ∀ i, x ∈ K i → ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, closedBall p.2 p.1 ⊆ U i by
apply mp_mem ((eventually_all_finite (hfin.point_finite x)).2 this)
(mp_mem (@tendsto_snd ℝ≥0∞ _ (𝓝 0) _ _ (hfin.iInter_compl_mem_nhds hK x)) _)
apply univ_mem'
rintro ⟨r, y⟩ hxy hyU i hi
simp only [mem_iInter... | 2,094 |
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal NNReal Filter Set Fu... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 64 | 72 | theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
(hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
∃ r : ℝ, ∀ᶠ y in 𝓝 x,
r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i } := by |
have := (ENNReal.continuous_ofReal.tendsto' 0 0 ENNReal.ofReal_zero).eventually
(eventually_nhds_zero_forall_closedBall_subset hK hU hKU hfin x).curry
rcases this.exists_gt with ⟨r, hr0, hr⟩
refine ⟨r, hr.mono fun y hy => ⟨hr0, ?_⟩⟩
rwa [mem_preimage, mem_iInter₂]
| 2,094 |
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal NNReal Filter Set Fu... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 87 | 93 | theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧
∀ (i), ∀ x ∈ K i, closedBall x (ENNReal.ofReal <| δ x) ⊆ U i := by |
simpa only [mem_inter_iff, forall_and, mem_preimage, mem_iInter, @forall_swap ι X] using
exists_continuous_forall_mem_convex_of_local_const exists_forall_closedBall_subset_aux₂
(exists_forall_closedBall_subset_aux₁ hK hU hKU hfin)
| 2,094 |
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal NNReal Filter Set Fu... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 100 | 106 | theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i := by |
rcases exists_continuous_real_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ₀, hδ⟩
lift δ to C(X, ℝ≥0) using fun x => (hδ₀ x).le
refine ⟨δ, hδ₀, fun i x hi => ?_⟩
simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi
| 2,094 |
import Mathlib.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import Mathlib.Topology.ShrinkingLemma
#align_import geometry.manifold.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 157 | 162 | theorem exists_pos_of_mem {x} (hx : x ∈ s) : ∃ i, 0 < f i x := by |
by_contra! h
have H : ∀ i, f i x = 0 := fun i ↦ le_antisymm (h i) (f.nonneg i x)
have := f.sum_eq_one hx
simp_rw [H] at this
simpa
| 2,095 |
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 68 | 75 | theorem embeddingPiTangent_injOn : InjOn f.embeddingPiTangent s := by |
intro x hx y _ h
simp only [embeddingPiTangent_coe, funext_iff] at h
obtain ⟨h₁, h₂⟩ := Prod.mk.inj_iff.1 (h (f.ind x hx))
rw [f.apply_ind x hx] at h₂
rw [← h₂, f.apply_ind x hx, one_smul, one_smul] at h₁
have := f.mem_extChartAt_source_of_eq_one h₂.symm
exact (extChartAt I (f.c _)).injOn (f.mem_extChart... | 2,096 |
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 83 | 98 | theorem comp_embeddingPiTangent_mfderiv (x : M) (hx : x ∈ s) :
((ContinuousLinearMap.fst ℝ E ℝ).comp
(@ContinuousLinearMap.proj ℝ _ ι (fun _ => E × ℝ) _ _ (fun _ => inferInstance)
(f.ind x hx))).comp
(mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) =
mfderiv I I (chartAt H... |
set L :=
(ContinuousLinearMap.fst ℝ E ℝ).comp
(@ContinuousLinearMap.proj ℝ _ ι (fun _ => E × ℝ) _ _ (fun _ => inferInstance) (f.ind x hx))
have := L.hasMFDerivAt.comp x f.embeddingPiTangent.smooth.mdifferentiableAt.hasMFDerivAt
convert hasMFDerivAt_unique this _
refine (hasMFDerivAt_extChartAt I (f.m... | 2,096 |
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 101 | 107 | theorem embeddingPiTangent_ker_mfderiv (x : M) (hx : x ∈ s) :
LinearMap.ker (mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) = ⊥ := by |
apply bot_unique
rw [← (mdifferentiable_chart I (f.c (f.ind x hx))).ker_mfderiv_eq_bot
(f.mem_chartAt_ind_source x hx),
← comp_embeddingPiTangent_mfderiv]
exact LinearMap.ker_le_ker_comp _ _
| 2,096 |
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 118 | 133 | theorem exists_immersion_euclidean [Finite ι] (f : SmoothBumpCovering ι I M) :
∃ (n : ℕ) (e : M → EuclideanSpace ℝ (Fin n)),
Smooth I (𝓡 n) e ∧ Injective e ∧ ∀ x : M, Injective (mfderiv I (𝓡 n) e x) := by |
cases nonempty_fintype ι
set F := EuclideanSpace ℝ (Fin <| finrank ℝ (ι → E × ℝ))
letI : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.2 inferInstance
letI : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.1 inferInstance
set eEF : (ι → E × ℝ) ≃L[ℝ] F :=
ContinuousLinearEquiv.ofFinrankEq finrank_e... | 2,096 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 65 | 69 | theorem Subsingleton.convexIndependent [Subsingleton ι] (p : ι → E) : ConvexIndependent 𝕜 p := by |
intro s x hx
have : (convexHull 𝕜 (p '' s)).Nonempty := ⟨p x, hx⟩
rw [convexHull_nonempty_iff, Set.image_nonempty] at this
rwa [Subsingleton.mem_iff_nonempty]
| 2,097 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 82 | 86 | theorem ConvexIndependent.comp_embedding {ι' : Type*} (f : ι' ↪ ι) {p : ι → E}
(hc : ConvexIndependent 𝕜 p) : ConvexIndependent 𝕜 (p ∘ f) := by |
intro s x hx
rw [← f.injective.mem_set_image]
exact hc _ _ (by rwa [Set.image_image])
| 2,097 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 133 | 141 | theorem convexIndependent_iff_not_mem_convexHull_diff {p : ι → E} :
ConvexIndependent 𝕜 p ↔ ∀ i s, p i ∉ convexHull 𝕜 (p '' (s \ {i})) := by |
refine ⟨fun hc i s h => ?_, fun h s i hi => ?_⟩
· rw [hc.mem_convexHull_iff] at h
exact h.2 (Set.mem_singleton _)
· by_contra H
refine h i s ?_
rw [Set.diff_singleton_eq_self H]
exact hi
| 2,097 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 144 | 153 | theorem convexIndependent_set_iff_inter_convexHull_subset {s : Set E} :
ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ t, t ⊆ s → s ∩ convexHull 𝕜 t ⊆ t := by |
constructor
· rintro hc t h x ⟨hxs, hxt⟩
refine hc { x | ↑x ∈ t } ⟨x, hxs⟩ ?_
rw [Subtype.coe_image_of_subset h]
exact hxt
· intro hc t x h
rw [← Subtype.coe_injective.mem_set_image]
exact hc (t.image ((↑) : s → E)) (Subtype.coe_image_subset s t) ⟨x.prop, h⟩
| 2,097 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 158 | 166 | theorem convexIndependent_set_iff_not_mem_convexHull_diff {s : Set E} :
ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ x ∈ s, x ∉ convexHull 𝕜 (s \ {x}) := by |
rw [convexIndependent_set_iff_inter_convexHull_subset]
constructor
· rintro hs x hxs hx
exact (hs _ Set.diff_subset ⟨hxs, hx⟩).2 (Set.mem_singleton _)
· rintro hs t ht x ⟨hxs, hxt⟩
by_contra h
exact hs _ hxs (convexHull_mono (Set.subset_diff_singleton ht h) hxt)
| 2,097 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Join
#align_import analysis.convex.stone_separation from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
variable {𝕜 E ι : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
| Mathlib/Analysis/Convex/StoneSeparation.lean | 30 | 77 | theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
(hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) := by |
rw [not_disjoint_iff]
obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
obtain rfl | haz' := haz.eq_or_lt
· rw [zero_add] at habz
rw [zero_smul, zero_add, habz, one_smul]
refine ⟨v, by apply right_mem_segment, segment_subset_convexHull ?_ ?_ hv⟩ <;> simp
obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
obtain r... | 2,098 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Join
#align_import analysis.convex.stone_separation from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
variable {𝕜 E ι : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
th... | Mathlib/Analysis/Convex/StoneSeparation.lean | 81 | 109 | theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by |
let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
obtain ⟨C, hC, hsC, hCmax⟩ :=
zorn_subset_nonempty S
(fun c hcS hc ⟨_, _⟩ =>
⟨⋃₀ c,
⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
fun s => subset_sUnion... | 2,098 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.jensen from "leanprover-community/mathlib"@"bfad3f455b388fbcc14c49d0cac884f774f14d20"
open Finset LinearMap Set
open scoped Classical
open Convex Pointwise
variable {�... | Mathlib/Analysis/Convex/Jensen.lean | 52 | 58 | theorem ConvexOn.map_centerMass_le (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
(h₁ : 0 < ∑ i ∈ t, w i) (hmem : ∀ i ∈ t, p i ∈ s) :
f (t.centerMass w p) ≤ t.centerMass w (f ∘ p) := by |
have hmem' : ∀ i ∈ t, (p i, (f ∘ p) i) ∈ { p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2 } := fun i hi =>
⟨hmem i hi, le_rfl⟩
convert (hf.convex_epigraph.centerMass_mem h₀ h₁ hmem').2 <;>
simp only [centerMass, Function.comp, Prod.smul_fst, Prod.fst_sum, Prod.smul_snd, Prod.snd_sum]
| 2,099 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.jensen from "leanprover-community/mathlib"@"bfad3f455b388fbcc14c49d0cac884f774f14d20"
open Finset LinearMap Set
open scoped Classical
open Convex Pointwise
variable {�... | Mathlib/Analysis/Convex/Jensen.lean | 69 | 72 | theorem ConvexOn.map_sum_le (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1)
(hmem : ∀ i ∈ t, p i ∈ s) : f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i) := by |
simpa only [centerMass, h₁, inv_one, one_smul] using
hf.map_centerMass_le h₀ (h₁.symm ▸ zero_lt_one) hmem
| 2,099 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.mean_inequalities_pow from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
universe u... | Mathlib/Analysis/MeanInequalitiesPow.lean | 72 | 86 | theorem pow_sum_div_card_le_sum_pow {f : ι → ℝ} (n : ℕ) (hf : ∀ a ∈ s, 0 ≤ f a) :
(∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) := by |
rcases s.eq_empty_or_nonempty with (rfl | hs)
· simp_rw [Finset.sum_empty, zero_pow n.succ_ne_zero, zero_div]; rfl
· have hs0 : 0 < (s.card : ℝ) := Nat.cast_pos.2 hs.card_pos
suffices (∑ x ∈ s, f x / s.card) ^ (n + 1) ≤ ∑ x ∈ s, f x ^ (n + 1) / s.card by
rwa [← Finset.sum_div, ← Finset.sum_div, div_pow... | 2,100 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.mean_inequalities_pow from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
universe u... | Mathlib/Analysis/MeanInequalitiesPow.lean | 101 | 110 | theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1)
(hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) := by |
have : 0 < p := by positivity
rw [← rpow_le_rpow_iff _ _ this, ← rpow_mul, one_div_mul_cancel (ne_of_gt this), rpow_one]
· exact rpow_arith_mean_le_arith_mean_rpow s w z hw hw' hz hp
all_goals
apply_rules [sum_nonneg, rpow_nonneg]
intro i hi
apply_rules [mul_nonneg, rpow_nonneg, hw i hi, hz i hi]
| 2,100 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 113 | 134 | theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i ∈ s, z i ^ w i ≤ ∑ i ∈ s, w i * z i := by |
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- I... | 2,101 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 138 | 148 | theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i ∈ s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹ ≤ (∑ i ∈ s, w i * z i) / (∑ i ∈ s, w i) := by |
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i ∈ s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_a... | 2,101 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 150 | 166 | theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i ∈ s, z i ^ w i = x :=
calc
∏ i ∈ s, z i ^ w i = ∏ i ∈ s, x ^ w i := by |
refine prod_congr rfl fun i hi => ?_
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i ∈ s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, ... | 2,101 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 169 | 177 | theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i ∈ s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i ∈ s, w i * z i = x :=
calc
∑ i ∈ s, w i * z i = ∑ i ∈ s, w i * x := by |
refine sum_congr rfl fun i hi => ?_
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
| 2,101 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 180 | 183 | theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i := by |
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
| 2,101 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section EDist
variable [EDist α] [EDist β]
open scope... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 161 | 164 | theorem prod_edist_eq_card (f g : WithLp 0 (α × β)) :
edist f g =
(if edist f.fst g.fst = 0 then 0 else 1) + (if edist f.snd g.snd = 0 then 0 else 1) := by |
convert if_pos rfl
| 2,102 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section EDist
variable [EDist α] [EDist β]
open scope... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 171 | 174 | theorem prod_edist_eq_sup (f g : WithLp ∞ (α × β)) :
edist f g = edist f.fst g.fst ⊔ edist f.snd g.snd := by |
dsimp [edist]
exact if_neg ENNReal.top_ne_zero
| 2,102 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section Dist
variable [Dist α] [Dist β]
open scoped C... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 231 | 233 | theorem prod_dist_eq_card (f g : WithLp 0 (α × β)) : dist f g =
(if dist f.fst g.fst = 0 then 0 else 1) + (if dist f.snd g.snd = 0 then 0 else 1) := by |
convert if_pos rfl
| 2,102 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section Dist
variable [Dist α] [Dist β]
open scoped C... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 240 | 243 | theorem prod_dist_eq_sup (f g : WithLp ∞ (α × β)) :
dist f g = dist f.fst g.fst ⊔ dist f.snd g.snd := by |
dsimp [dist]
exact if_neg ENNReal.top_ne_zero
| 2,102 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section Norm
variable [Norm α] [Norm β]
open scoped C... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 270 | 272 | theorem prod_norm_eq_card (f : WithLp 0 (α × β)) :
‖f‖ = (if ‖f.fst‖ = 0 then 0 else 1) + (if ‖f.snd‖ = 0 then 0 else 1) := by |
convert if_pos rfl
| 2,102 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section Norm
variable [Norm α] [Norm β]
open scoped C... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 274 | 276 | theorem prod_norm_eq_sup (f : WithLp ∞ (α × β)) : ‖f‖ = ‖f.fst‖ ⊔ ‖f.snd‖ := by |
dsimp [Norm.norm]
exact if_neg ENNReal.top_ne_zero
| 2,102 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.NormedSpace.WithLp
#align_import analysis.normed_space.pi_Lp from "leanprover-community/mathlib"@"9d013ad8430ddddd350cff5c3db830278ded3c79"
set_option linter.uppercaseLean3 f... | Mathlib/Analysis/NormedSpace/PiLp.lean | 185 | 187 | theorem edist_eq_iSup (f g : PiLp ∞ β) : edist f g = ⨆ i, edist (f i) (g i) := by |
dsimp [edist]
exact if_neg ENNReal.top_ne_zero
| 2,103 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.NormedSpace.WithLp
#align_import analysis.normed_space.pi_Lp from "leanprover-community/mathlib"@"9d013ad8430ddddd350cff5c3db830278ded3c79"
set_option linter.uppercaseLean3 f... | Mathlib/Analysis/NormedSpace/PiLp.lean | 247 | 249 | theorem dist_eq_iSup (f g : PiLp ∞ α) : dist f g = ⨆ i, dist (f i) (g i) := by |
dsimp [dist]
exact if_neg ENNReal.top_ne_zero
| 2,103 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.NormedSpace.WithLp
#align_import analysis.normed_space.pi_Lp from "leanprover-community/mathlib"@"9d013ad8430ddddd350cff5c3db830278ded3c79"
set_option linter.uppercaseLean3 f... | Mathlib/Analysis/NormedSpace/PiLp.lean | 276 | 278 | theorem norm_eq_ciSup (f : PiLp ∞ β) : ‖f‖ = ⨆ i, ‖f i‖ := by |
dsimp [Norm.norm]
exact if_neg ENNReal.top_ne_zero
| 2,103 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 81 | 83 | theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by |
dsimp [Memℓp]
rw [if_pos rfl]
| 2,104 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 90 | 92 | theorem memℓp_infty_iff {f : ∀ i, E i} : Memℓp f ∞ ↔ BddAbove (Set.range fun i => ‖f i‖) := by |
dsimp [Memℓp]
rw [if_neg ENNReal.top_ne_zero, if_pos rfl]
| 2,104 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 99 | 103 | theorem memℓp_gen_iff (hp : 0 < p.toReal) {f : ∀ i, E i} :
Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal := by |
rw [ENNReal.toReal_pos_iff] at hp
dsimp [Memℓp]
rw [if_neg hp.1.ne', if_neg hp.2.ne]
| 2,104 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 106 | 114 | theorem memℓp_gen {f : ∀ i, E i} (hf : Summable fun i => ‖f i‖ ^ p.toReal) : Memℓp f p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
exact (Set.Finite.of_summable_const (by norm_num) H).subset (Set.subset_univ _)
· apply memℓp_infty
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
simpa using ((Se... | 2,104 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 117 | 127 | theorem memℓp_gen' {C : ℝ} {f : ∀ i, E i} (hf : ∀ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C) :
Memℓp f p := by |
apply memℓp_gen
use ⨆ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal
apply hasSum_of_isLUB_of_nonneg
· intro b
exact Real.rpow_nonneg (norm_nonneg _) _
apply isLUB_ciSup
use C
rintro - ⟨s, rfl⟩
exact hf s
| 2,104 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 130 | 138 | theorem zero_memℓp : Memℓp (0 : ∀ i, E i) p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp
· apply memℓp_infty
simp only [norm_zero, Pi.zero_apply]
exact bddAbove_singleton.mono Set.range_const_subset
· apply memℓp_gen
simp [Real.zero_rpow hp.ne', summable_zero]
| 2,104 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 160 | 167 | theorem neg {f : ∀ i, E i} (hf : Memℓp f p) : Memℓp (-f) p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp
| 2,104 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 175 | 211 | theorem of_exponent_ge {p q : ℝ≥0∞} {f : ∀ i, E i} (hfq : Memℓp f q) (hpq : q ≤ p) : Memℓp f p := by |
rcases ENNReal.trichotomy₂ hpq with
(⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ | ⟨rfl, hp⟩ | ⟨rfl, rfl⟩ | ⟨hq, rfl⟩ | ⟨hq, _, hpq'⟩)
· exact hfq
· apply memℓp_infty
obtain ⟨C, hC⟩ := (hfq.finite_dsupport.image fun i => ‖f i‖).bddAbove
use max 0 C
rintro x ⟨i, rfl⟩
by_cases hi : f i = 0
· simp [hi]
· ex... | 2,104 |
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.kuratowski from "leanprover-community/mathlib"@"95d4f6586d313c8c28e00f36621d2a6a66893aa6"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Metric TopologicalSpace NNReal ENNR... | Mathlib/Topology/MetricSpace/Kuratowski.lean | 52 | 57 | theorem embeddingOfSubset_dist_le (a b : α) :
dist (embeddingOfSubset x a) (embeddingOfSubset x b) ≤ dist a b := by |
refine lp.norm_le_of_forall_le dist_nonneg fun n => ?_
simp only [lp.coeFn_sub, Pi.sub_apply, embeddingOfSubset_coe, Real.dist_eq]
convert abs_dist_sub_le a b (x n) using 2
ring
| 2,105 |
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.kuratowski from "leanprover-community/mathlib"@"95d4f6586d313c8c28e00f36621d2a6a66893aa6"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Metric TopologicalSpace NNReal ENNR... | Mathlib/Topology/MetricSpace/Kuratowski.lean | 61 | 87 | theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x) := by |
refine Isometry.of_dist_eq fun a b => ?_
refine (embeddingOfSubset_dist_le x a b).antisymm (le_of_forall_pos_le_add fun e epos => ?_)
-- First step: find n with dist a (x n) < e
rcases Metric.mem_closure_range_iff.1 (H a) (e / 2) (half_pos epos) with ⟨n, hn⟩
-- Second step: use the norm control at index n to... | 2,105 |
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.kuratowski from "leanprover-community/mathlib"@"95d4f6586d313c8c28e00f36621d2a6a66893aa6"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Metric TopologicalSpace NNReal ENNR... | Mathlib/Topology/MetricSpace/Kuratowski.lean | 91 | 102 | theorem exists_isometric_embedding (α : Type u) [MetricSpace α] [SeparableSpace α] :
∃ f : α → ℓ^∞(ℕ), Isometry f := by |
rcases (univ : Set α).eq_empty_or_nonempty with h | h
· use fun _ => 0; intro x; exact absurd h (Nonempty.ne_empty ⟨x, mem_univ x⟩)
· -- We construct a map x : ℕ → α with dense image
rcases h with ⟨basepoint⟩
haveI : Inhabited α := ⟨basepoint⟩
have : ∃ s : Set α, s.Countable ∧ Dense s := exists_count... | 2,105 |
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Topology.ContinuousFunction.Bounded
#align_import analysis.normed_space.lp_equiv from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open scoped ENNReal
section LpPiLp
set_option linter... | Mathlib/Analysis/NormedSpace/LpEquiv.lean | 54 | 58 | theorem Memℓp.all (f : ∀ i, E i) : Memℓp f p := by |
rcases p.trichotomy with (rfl | rfl | _h)
· exact memℓp_zero_iff.mpr { i : α | f i ≠ 0 }.toFinite
· exact memℓp_infty_iff.mpr (Set.Finite.bddAbove (Set.range fun i : α ↦ ‖f i‖).toFinite)
· cases nonempty_fintype α; exact memℓp_gen ⟨Finset.univ.sum _, hasSum_fintype _⟩
| 2,106 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 66 | 79 | theorem lintegral_mul_le_one_of_lintegral_rpow_eq_one {p q : ℝ} (hpq : p.IsConjExponent q)
{f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_norm : ∫⁻ a, f a ^ p ∂μ = 1)
(hg_norm : ∫⁻ a, g a ^ q ∂μ = 1) : (∫⁻ a, (f * g) a ∂μ) ≤ 1 := by |
calc
(∫⁻ a : α, (f * g) a ∂μ) ≤
∫⁻ a : α, f a ^ p / ENNReal.ofReal p + g a ^ q / ENNReal.ofReal q ∂μ :=
lintegral_mono fun a => young_inequality (f a) (g a) hpq
_ = 1 := by
simp only [div_eq_mul_inv]
rw [lintegral_add_left']
· rw [lintegral_mul_const'' _ (hf.pow_const p), lint... | 2,107 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 87 | 90 | theorem fun_eq_funMulInvSnorm_mul_snorm {p : ℝ} (f : α → ℝ≥0∞) (hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0)
(hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) {a : α} :
f a = funMulInvSnorm f p μ a * (∫⁻ c, f c ^ p ∂μ) ^ (1 / p) := by |
simp [funMulInvSnorm, mul_assoc, ENNReal.inv_mul_cancel, hf_nonzero, hf_top]
| 2,107 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 93 | 98 | theorem funMulInvSnorm_rpow {p : ℝ} (hp0 : 0 < p) {f : α → ℝ≥0∞} {a : α} :
funMulInvSnorm f p μ a ^ p = f a ^ p * (∫⁻ c, f c ^ p ∂μ)⁻¹ := by |
rw [funMulInvSnorm, mul_rpow_of_nonneg _ _ (le_of_lt hp0)]
suffices h_inv_rpow : ((∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ c : α, f c ^ p ∂μ)⁻¹ by
rw [h_inv_rpow]
rw [inv_rpow, ← rpow_mul, one_div_mul_cancel hp0.ne', rpow_one]
| 2,107 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 101 | 106 | theorem lintegral_rpow_funMulInvSnorm_eq_one {p : ℝ} (hp0_lt : 0 < p) {f : α → ℝ≥0∞}
(hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) :
∫⁻ c, funMulInvSnorm f p μ c ^ p ∂μ = 1 := by |
simp_rw [funMulInvSnorm_rpow hp0_lt]
rw [lintegral_mul_const', ENNReal.mul_inv_cancel hf_nonzero hf_top]
rwa [inv_ne_top]
| 2,107 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 110 | 130 | theorem lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top {p q : ℝ} (hpq : p.IsConjExponent q)
{f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_nontop : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤)
(hg_nontop : (∫⁻ a, g a ^ q ∂μ) ≠ ⊤) (hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0)
(hg_nonzero : (∫⁻ a, g a ^ q ∂μ) ≠ 0) :
(∫⁻ a, (f * g) a ∂... |
let npf := (∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p)
let nqg := (∫⁻ c : α, g c ^ q ∂μ) ^ (1 / q)
calc
(∫⁻ a : α, (f * g) a ∂μ) =
∫⁻ a : α, (funMulInvSnorm f p μ * funMulInvSnorm g q μ) a * (npf * nqg) ∂μ := by
refine lintegral_congr fun a => ?_
rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_snorm f h... | 2,107 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 26 | 33 | theorem snorm'_add_le {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
(hq1 : 1 ≤ q) : snorm' (f + g) q μ ≤ snorm' f q μ + snorm' g q μ :=
calc
(∫⁻ a, (‖(f + g) a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) ≤
(∫⁻ a, ((fun a => (‖f a‖₊ : ℝ≥0∞)) + fun a => (‖g a‖₊ : ℝ≥0∞)) a ^ q ∂μ) ^ (1 / ... |
gcongr with a
simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]
_ ≤ snorm' f q μ + snorm' g q μ := ENNReal.lintegral_Lp_add_le hf.ennnorm hg.ennnorm hq1
| 2,108 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 36 | 44 | theorem snorm'_add_le_of_le_one {f g : α → E} (hf : AEStronglyMeasurable f μ) (hq0 : 0 ≤ q)
(hq1 : q ≤ 1) : snorm' (f + g) q μ ≤ (2 : ℝ≥0∞) ^ (1 / q - 1) * (snorm' f q μ + snorm' g q μ) :=
calc
(∫⁻ a, (‖(f + g) a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) ≤
(∫⁻ a, ((fun a => (‖f a‖₊ : ℝ≥0∞)) + fun a => (‖g a‖₊ : ℝ≥0... |
gcongr with a
simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]
_ ≤ (2 : ℝ≥0∞) ^ (1 / q - 1) * (snorm' f q μ + snorm' g q μ) :=
ENNReal.lintegral_Lp_add_le_of_le_one hf.ennnorm hq0 hq1
| 2,108 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 47 | 51 | theorem snormEssSup_add_le {f g : α → E} :
snormEssSup (f + g) μ ≤ snormEssSup f μ + snormEssSup g μ := by |
refine le_trans (essSup_mono_ae (eventually_of_forall fun x => ?_)) (ENNReal.essSup_add_le _ _)
simp_rw [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe]
exact nnnorm_add_le _ _
| 2,108 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 54 | 63 | theorem snorm_add_le {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
(hp1 : 1 ≤ p) : snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ := by |
by_cases hp0 : p = 0
· simp [hp0]
by_cases hp_top : p = ∞
· simp [hp_top, snormEssSup_add_le]
have hp1_real : 1 ≤ p.toReal := by
rwa [← ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top hp_top]
repeat rw [snorm_eq_snorm' hp0 hp_top]
exact snorm'_add_le hf hg hp1_real
| 2,108 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 73 | 76 | theorem LpAddConst_of_one_le {p : ℝ≥0∞} (hp : 1 ≤ p) : LpAddConst p = 1 := by |
rw [LpAddConst, if_neg]
intro h
exact lt_irrefl _ (h.2.trans_le hp)
| 2,108 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 80 | 83 | theorem LpAddConst_zero : LpAddConst 0 = 1 := by |
rw [LpAddConst, if_neg]
intro h
exact lt_irrefl _ h.1
| 2,108 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 87 | 94 | theorem LpAddConst_lt_top (p : ℝ≥0∞) : LpAddConst p < ∞ := by |
rw [LpAddConst]
split_ifs with h
· apply ENNReal.rpow_lt_top_of_nonneg _ ENNReal.two_ne_top
simp only [one_div, sub_nonneg]
apply one_le_inv (ENNReal.toReal_pos h.1.ne' (h.2.trans ENNReal.one_lt_top).ne)
simpa using ENNReal.toReal_mono ENNReal.one_ne_top h.2.le
· exact ENNReal.one_lt_top
| 2,108 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 98 | 109 | theorem snorm_add_le' {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
(p : ℝ≥0∞) : snorm (f + g) p μ ≤ LpAddConst p * (snorm f p μ + snorm g p μ) := by |
rcases eq_or_ne p 0 with (rfl | hp)
· simp only [snorm_exponent_zero, add_zero, mul_zero, le_zero_iff]
rcases lt_or_le p 1 with (h'p | h'p)
· simp only [snorm_eq_snorm' hp (h'p.trans ENNReal.one_lt_top).ne]
convert snorm'_add_le_of_le_one hf ENNReal.toReal_nonneg _
· have : p ∈ Set.Ioo (0 : ℝ≥0∞) 1 := ... | 2,108 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 26 | 45 | theorem snorm'_le_snorm'_mul_rpow_measure_univ {p q : ℝ} (hp0_lt : 0 < p) (hpq : p ≤ q)
(hf : AEStronglyMeasurable f μ) :
snorm' f p μ ≤ snorm' f q μ * μ Set.univ ^ (1 / p - 1 / q) := by |
have hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq
by_cases hpq_eq : p = q
· rw [hpq_eq, sub_self, ENNReal.rpow_zero, mul_one]
have hpq : p < q := lt_of_le_of_ne hpq hpq_eq
let g := fun _ : α => (1 : ℝ≥0∞)
have h_rw : (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ p ∂μ) = ∫⁻ a, ((‖f a‖₊ : ℝ≥0∞) * g a) ^ p ∂μ :=
lintegral_cong... | 2,109 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 48 | 58 | theorem snorm'_le_snormEssSup_mul_rpow_measure_univ {q : ℝ} (hq_pos : 0 < q) :
snorm' f q μ ≤ snormEssSup f μ * μ Set.univ ^ (1 / q) := by |
have h_le : (∫⁻ a : α, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) ≤ ∫⁻ _ : α, snormEssSup f μ ^ q ∂μ := by
refine lintegral_mono_ae ?_
have h_nnnorm_le_snorm_ess_sup := coe_nnnorm_ae_le_snormEssSup f μ
exact h_nnnorm_le_snorm_ess_sup.mono fun x hx => by gcongr
rw [snorm', ← ENNReal.rpow_one (snormEssSup f μ)]
nth_rw 2 ... | 2,109 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 61 | 85 | theorem snorm_le_snorm_mul_rpow_measure_univ {p q : ℝ≥0∞} (hpq : p ≤ q)
(hf : AEStronglyMeasurable f μ) :
snorm f p μ ≤ snorm f q μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal) := by |
by_cases hp0 : p = 0
· simp [hp0, zero_le]
rw [← Ne] at hp0
have hp0_lt : 0 < p := lt_of_le_of_ne (zero_le _) hp0.symm
have hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq
by_cases hq_top : q = ∞
· simp only [hq_top, _root_.div_zero, one_div, ENNReal.top_toReal, sub_zero, snorm_exponent_top,
GroupWithZ... | 2,109 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 88 | 92 | theorem snorm'_le_snorm'_of_exponent_le {p q : ℝ} (hp0_lt : 0 < p)
(hpq : p ≤ q) (μ : Measure α) [IsProbabilityMeasure μ] (hf : AEStronglyMeasurable f μ) :
snorm' f p μ ≤ snorm' f q μ := by |
have h_le_μ := snorm'_le_snorm'_mul_rpow_measure_univ hp0_lt hpq hf
rwa [measure_univ, ENNReal.one_rpow, mul_one] at h_le_μ
| 2,109 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section B... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 158 | 196 | theorem snorm_le_snorm_top_mul_snorm (p : ℝ≥0∞) (f : α → E) {g : α → F}
(hg : AEStronglyMeasurable g μ) (b : E → F → G)
(h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) :
snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ∞ μ * snorm g p μ := by |
by_cases hp_top : p = ∞
· simp_rw [hp_top, snorm_exponent_top]
refine le_trans (essSup_mono_ae <| h.mono fun a ha => ?_) (ENNReal.essSup_mul_le _ _)
simp_rw [Pi.mul_apply, ← ENNReal.coe_mul, ENNReal.coe_le_coe]
exact ha
by_cases hp_zero : p = 0
· simp only [hp_zero, snorm_exponent_zero, mul_zero, l... | 2,109 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section B... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 208 | 222 | theorem snorm'_le_snorm'_mul_snorm' {p q r : ℝ} (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) (b : E → F → G)
(h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) (hp0_lt : 0 < p) (hpq : p < q)
(hpqr : 1 / p = 1 / q + 1 / r) :
snorm' (fun x => b (f x) (g x)) p μ ≤ snorm' f q μ * snorm' g ... |
rw [snorm']
calc
(∫⁻ a : α, ↑‖b (f a) (g a)‖₊ ^ p ∂μ) ^ (1 / p) ≤
(∫⁻ a : α, ↑(‖f a‖₊ * ‖g a‖₊) ^ p ∂μ) ^ (1 / p) :=
(ENNReal.rpow_le_rpow_iff <| one_div_pos.mpr hp0_lt).mpr <|
lintegral_mono_ae <|
h.mono fun a ha => (ENNReal.rpow_le_rpow_iff hp0_lt).mpr <| ENNReal.coe_le_coe.mp... | 2,109 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 95 | 97 | theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by | simp [hfp.2]
| 2,110 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 126 | 127 | theorem toLp_congr {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) (hfg : f =ᵐ[μ] g) :
hf.toLp f = hg.toLp g := by | simp [toLp, hfg]
| 2,110 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 131 | 132 | theorem toLp_eq_toLp_iff {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
hf.toLp f = hg.toLp g ↔ f =ᵐ[μ] g := by | simp [toLp]
| 2,110 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 163 | 167 | theorem ext {f g : Lp E p μ} (h : f =ᵐ[μ] g) : f = g := by |
cases f
cases g
simp only [Subtype.mk_eq_mk]
exact AEEqFun.ext h
| 2,110 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 177 | 178 | theorem mem_Lp_iff_memℒp {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ Memℒp f p μ := by |
simp [mem_Lp_iff_snorm_lt_top, Memℒp, f.stronglyMeasurable.aestronglyMeasurable]
| 2,110 |
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.lp_order from "leanprover-community/mathlib"@"5dc275ec639221ca4d5f56938eb966f6ad9bc89f"
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory
open scoped ENNReal
... | Mathlib/MeasureTheory/Function/LpOrder.lean | 41 | 42 | theorem coeFn_le (f g : Lp E p μ) : f ≤ᵐ[μ] g ↔ f ≤ g := by |
rw [← Subtype.coe_le_coe, ← AEEqFun.coeFn_le]
| 2,111 |
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.lp_order from "leanprover-community/mathlib"@"5dc275ec639221ca4d5f56938eb966f6ad9bc89f"
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory
open scoped ENNReal
... | Mathlib/MeasureTheory/Function/LpOrder.lean | 45 | 50 | theorem coeFn_nonneg (f : Lp E p μ) : 0 ≤ᵐ[μ] f ↔ 0 ≤ f := by |
rw [← coeFn_le]
have h0 := Lp.coeFn_zero E p μ
constructor <;> intro h <;> filter_upwards [h, h0] with _ _ h2
· rwa [h2]
· rwa [← h2]
| 2,111 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 66 | 67 | theorem lintegral_nnnorm_eq_lintegral_edist (f : α → β) :
∫⁻ a, ‖f a‖₊ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by | simp only [edist_eq_coe_nnnorm]
| 2,112 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 70 | 72 | theorem lintegral_norm_eq_lintegral_edist (f : α → β) :
∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by |
simp only [ofReal_norm_eq_coe_nnnorm, edist_eq_coe_nnnorm]
| 2,112 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 75 | 80 | theorem lintegral_edist_triangle {f g h : α → β} (hf : AEStronglyMeasurable f μ)
(hh : AEStronglyMeasurable h μ) :
(∫⁻ a, edist (f a) (g a) ∂μ) ≤ (∫⁻ a, edist (f a) (h a) ∂μ) + ∫⁻ a, edist (g a) (h a) ∂μ := by |
rw [← lintegral_add_left' (hf.edist hh)]
refine lintegral_mono fun a => ?_
apply edist_triangle_right
| 2,112 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 83 | 83 | theorem lintegral_nnnorm_zero : (∫⁻ _ : α, ‖(0 : β)‖₊ ∂μ) = 0 := by | simp
| 2,112 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 96 | 97 | theorem lintegral_nnnorm_neg {f : α → β} : (∫⁻ a, ‖(-f) a‖₊ ∂μ) = ∫⁻ a, ‖f a‖₊ ∂μ := by |
simp only [Pi.neg_apply, nnnorm_neg]
| 2,112 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 113 | 115 | theorem hasFiniteIntegral_iff_norm (f : α → β) :
HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) < ∞ := by |
simp only [HasFiniteIntegral, ofReal_norm_eq_coe_nnnorm]
| 2,112 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 118 | 120 | theorem hasFiniteIntegral_iff_edist (f : α → β) :
HasFiniteIntegral f μ ↔ (∫⁻ a, edist (f a) 0 ∂μ) < ∞ := by |
simp only [hasFiniteIntegral_iff_norm, edist_dist, dist_zero_right]
| 2,112 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 123 | 125 | theorem hasFiniteIntegral_iff_ofReal {f : α → ℝ} (h : 0 ≤ᵐ[μ] f) :
HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal (f a) ∂μ) < ∞ := by |
rw [HasFiniteIntegral, lintegral_nnnorm_eq_of_ae_nonneg h]
| 2,112 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 128 | 130 | theorem hasFiniteIntegral_iff_ofNNReal {f : α → ℝ≥0} :
HasFiniteIntegral (fun x => (f x : ℝ)) μ ↔ (∫⁻ a, f a ∂μ) < ∞ := by |
simp [hasFiniteIntegral_iff_norm]
| 2,112 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 133 | 139 | theorem HasFiniteIntegral.mono {f : α → β} {g : α → γ} (hg : HasFiniteIntegral g μ)
(h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : HasFiniteIntegral f μ := by |
simp only [hasFiniteIntegral_iff_norm] at *
calc
(∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a : α, ENNReal.ofReal ‖g a‖ ∂μ :=
lintegral_mono_ae (h.mono fun a h => ofReal_le_ofReal h)
_ < ∞ := hg
| 2,112 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 99 | 99 | theorem integrableOn_empty : IntegrableOn f ∅ μ := by | simp [IntegrableOn, integrable_zero_measure]
| 2,113 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 103 | 104 | theorem integrableOn_univ : IntegrableOn f univ μ ↔ Integrable f μ := by |
rw [IntegrableOn, Measure.restrict_univ]
| 2,113 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 572 | 583 | theorem ContinuousOn.aemeasurable [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
[TopologicalSpace β] [BorelSpace β] {f : α → β} {s : Set α} {μ : Measure α}
(hf : ContinuousOn f s) (hs : MeasurableSet s) : AEMeasurable f (μ.restrict s) := by |
nontriviality α; inhabit α
have : (Set.piecewise s f fun _ => f default) =ᵐ[μ.restrict s] f := piecewise_ae_eq_restrict hs
refine ⟨Set.piecewise s f fun _ => f default, ?_, this.symm⟩
apply measurable_of_isOpen
intro t ht
obtain ⟨u, u_open, hu⟩ : ∃ u : Set α, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s :=
_root_.co... | 2,113 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 36 | 44 | theorem _root_.Asymptotics.IsBigO.integrableAtFilter [IsMeasurablyGenerated l]
(hf : f =O[l] g) (hfm : StronglyMeasurableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
IntegrableAtFilter f l μ := by |
obtain ⟨C, hC⟩ := hf.bound
obtain ⟨s, hsl, hsm, hfg, hf, hg⟩ :=
(hC.smallSets.and <| hfm.eventually.and hg.eventually).exists_measurable_mem_of_smallSets
refine ⟨s, hsl, (hg.norm.const_mul C).mono hf ?_⟩
refine (ae_restrict_mem hsm).mono fun x hx ↦ ?_
exact (hfg x hx).trans (le_abs_self _)
| 2,114 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 47 | 50 | theorem _root_.Asymptotics.IsBigO.integrable (hfm : AEStronglyMeasurable f μ)
(hf : f =O[⊤] g) (hg : Integrable g μ) : Integrable f μ := by |
rewrite [← integrableAtFilter_top] at *
exact hf.integrableAtFilter ⟨univ, univ_mem, hfm.restrict⟩ hg
| 2,114 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 58 | 62 | theorem LocallyIntegrable.integrable_of_isBigO_cocompact [IsMeasurablyGenerated (cocompact α)]
(hf : LocallyIntegrable f μ) (ho : f =O[cocompact α] g)
(hg : IntegrableAtFilter g (cocompact α) μ) : Integrable f μ := by |
refine integrable_iff_integrableAtFilter_cocompact.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter
| 2,114 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 70 | 77 | theorem LocallyIntegrable.integrable_of_isBigO_atBot_atTop
[IsMeasurablyGenerated (atBot (α := α))] [IsMeasurablyGenerated (atTop (α := α))]
(hf : LocallyIntegrable f μ)
(ho : f =O[atBot] g) (hg : IntegrableAtFilter g atBot μ)
(ho' : f =O[atTop] g') (hg' : IntegrableAtFilter g' atTop μ) : Integrable f μ... |
refine integrable_iff_integrableAtFilter_atBot_atTop.mpr
⟨⟨ho.integrableAtFilter ?_ hg, ho'.integrableAtFilter ?_ hg'⟩, hf⟩
all_goals exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter
| 2,114 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 81 | 85 | theorem LocallyIntegrableOn.integrableOn_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))]
(hf : LocallyIntegrableOn f (Iic a) μ) (ho : f =O[atBot] g)
(hg : IntegrableAtFilter g atBot μ) : IntegrableOn f (Iic a) μ := by |
refine integrableOn_Iic_iff_integrableAtFilter_atBot.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
exact ⟨Iic a, Iic_mem_atBot a, hf.aestronglyMeasurable⟩
| 2,114 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 89 | 93 | theorem LocallyIntegrableOn.integrableOn_of_isBigO_atTop [IsMeasurablyGenerated (atTop (α := α))]
(hf : LocallyIntegrableOn f (Ici a) μ) (ho : f =O[atTop] g)
(hg : IntegrableAtFilter g atTop μ) : IntegrableOn f (Ici a) μ := by |
refine integrableOn_Ici_iff_integrableAtFilter_atTop.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
exact ⟨Ici a, Ici_mem_atTop a, hf.aestronglyMeasurable⟩
| 2,114 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 97 | 101 | theorem LocallyIntegrable.integrable_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))]
[OrderTop α] (hf : LocallyIntegrable f μ) (ho : f =O[atBot] g)
(hg : IntegrableAtFilter g atBot μ) : Integrable f μ := by |
refine integrable_iff_integrableAtFilter_atBot.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter
| 2,114 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 105 | 109 | theorem LocallyIntegrable.integrable_of_isBigO_atTop [IsMeasurablyGenerated (atTop (α := α))]
[OrderBot α] (hf : LocallyIntegrable f μ) (ho : f =O[atTop] g)
(hg : IntegrableAtFilter g atTop μ) : Integrable f μ := by |
refine integrable_iff_integrableAtFilter_atTop.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter
| 2,114 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 68 | 74 | theorem nnnorm_approxOn_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
{y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
‖approxOn f hf s y₀ h₀ n x - f x‖₊ ≤ ‖f x - y₀‖₊ := by |
have := edist_approxOn_le hf h₀ x n
rw [edist_comm y₀] at this
simp only [edist_nndist, nndist_eq_nnnorm] at this
exact mod_cast this
| 2,115 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 77 | 82 | theorem norm_approxOn_y₀_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
{y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
‖approxOn f hf s y₀ h₀ n x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ := by |
have := edist_approxOn_y0_le hf h₀ x n
repeat rw [edist_comm y₀, edist_eq_coe_nnnorm_sub] at this
exact mod_cast this
| 2,115 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 85 | 90 | theorem norm_approxOn_zero_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
(h₀ : (0 : E) ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
‖approxOn f hf s 0 h₀ n x‖ ≤ ‖f x‖ + ‖f x‖ := by |
have := edist_approxOn_y0_le hf h₀ x n
simp [edist_comm (0 : E), edist_eq_coe_nnnorm] at this
exact mod_cast this
| 2,115 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 93 | 135 | theorem tendsto_approxOn_Lp_snorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
{s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (hp_ne_top : p ≠ ∞) {μ : Measure β}
(hμ : ∀ᵐ x ∂μ, f x ∈ closure s) (hi : snorm (fun x => f x - y₀) p μ < ∞) :
Tendsto (fun n => snorm (⇑(approxOn f hf s y₀ h₀ n) ... |
by_cases hp_zero : p = 0
· simpa only [hp_zero, snorm_exponent_zero] using tendsto_const_nhds
have hp : 0 < p.toReal := toReal_pos hp_zero hp_ne_top
suffices
Tendsto (fun n => ∫⁻ x, (‖approxOn f hf s y₀ h₀ n x - f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) atTop
(𝓝 0) by
simp only [snorm_eq_lintegral_rpow_n... | 2,115 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 296 | 322 | theorem measure_preimage_lt_top_of_memℒp (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) (f : α →ₛ E)
(hf : Memℒp f p μ) (y : E) (hy_ne : y ≠ 0) : μ (f ⁻¹' {y}) < ∞ := by |
have hp_pos_real : 0 < p.toReal := ENNReal.toReal_pos hp_pos hp_ne_top
have hf_snorm := Memℒp.snorm_lt_top hf
rw [snorm_eq_snorm' hp_pos hp_ne_top, f.snorm'_eq, ←
@ENNReal.lt_rpow_one_div_iff _ _ (1 / p.toReal) (by simp [hp_pos_real]),
@ENNReal.top_rpow_of_pos (1 / (1 / p.toReal)) (by simp [hp_pos_real])... | 2,115 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 325 | 337 | theorem memℒp_of_finite_measure_preimage (p : ℝ≥0∞) {f : α →ₛ E}
(hf : ∀ y, y ≠ 0 → μ (f ⁻¹' {y}) < ∞) : Memℒp f p μ := by |
by_cases hp0 : p = 0
· rw [hp0, memℒp_zero_iff_aestronglyMeasurable]; exact f.aestronglyMeasurable
by_cases hp_top : p = ∞
· rw [hp_top]; exact memℒp_top f μ
refine ⟨f.aestronglyMeasurable, ?_⟩
rw [snorm_eq_snorm' hp0 hp_top, f.snorm'_eq]
refine ENNReal.rpow_lt_top_of_nonneg (by simp) (ENNReal.sum_lt_top... | 2,115 |
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import measure_theory.function.strongly_measurable.lp from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter TopologicalSpace Function
op... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean | 40 | 54 | theorem Memℒp.finStronglyMeasurable_of_stronglyMeasurable (hf : Memℒp f p μ)
(hf_meas : StronglyMeasurable f) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
FinStronglyMeasurable f μ := by |
borelize G
haveI : SeparableSpace (Set.range f ∪ {0} : Set G) :=
hf_meas.separableSpace_range_union_singleton
let fs := SimpleFunc.approxOn f hf_meas.measurable (Set.range f ∪ {0}) 0 (by simp)
refine ⟨fs, ?_, ?_⟩
· have h_fs_Lp : ∀ n, Memℒp (fs n) p μ :=
SimpleFunc.memℒp_approxOn_range hf_meas.meas... | 2,116 |
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