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 |
|---|---|---|---|---|---|
23,921 | lean_file_chunk | ace22e3ae25e809cd5a6a5b74e36e4005b285ba1945038e2b137554eebd57f1c | lean_source_file | pkg_mathlib_source_file_chunks | rw [resultant_C_mul_right, inv_pow, inv_mul_eq_iff_eq_mul₀ (by simp [hg0])] at this
simpa [← hf.natDegree_eq_card_roots, inv_pow, mul_left_comm (_ ^ g.natDegree), hg0] using this
letI inst := hR.toField
let L := g.SplittingField
apply (algebraMap R L).injective
have := resultant_eq_prod_roots_sub (f.map (al... |
3,237 | lean_file_chunk | 40e09196b6dd33e609ce0fe1847be152ba195687b0d61935b431d54f7d25ed45 | lean_source_file | pkg_mathlib_source_file_chunks | (-1) ^ (f.natDegree * (f.natDegree - 1) / 2) * f.leadingCoeff * f.discr := by
rw [← natDegree_pos_iff_degree_pos] at hf
rw [resultant_comm, resultant, ← sylvesterDeriv_updateRow f hf, Matrix.det_updateRow_smul,
Matrix.updateRow_eq_self, discr, mul_comm f.natDegree]
ring_nf
rw [Nat.div_mul_cancel (by convert... |
2,960 | lean_file_chunk | eacea60514c84e79d29cd1bd780369d80dcf0a98424e28354678d9552daef731 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.RingTheory.Localization.Away.Basic
public import Mathlib.RingTheory.Localization.Submodule
/-!
# More lemmas on localization away
... |
2,381 | lean_file_chunk | 3949b1764023eb8bd7084940fbeef574d7fd4746c7bb44815ef0e27fbf55cb73 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.RingTheory.AdjoinRoot
public import Mathlib.RingTheory.Localization.Away.Basic
/-... |
23,949 | lean_file_chunk | be1c06bcb0135f7fd5cf55899583b2283d721b7c5c5efc54e50a32a3fb373cb4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.GroupTheory.MonoidLocalization.Away
public import Mathlib.Algebra.Algebra.Pi
publi... |
8,363 | lean_file_chunk | c7aa3180ab1ec386d3677f791abe468d070fce7c6804aeb86b1aba2dde7fbee3 | lean_source_file | pkg_mathlib_source_file_chunks | ∃! r : R, ∀ a : s, algebraMap R _ r = f a := by
have mem := (Ideal.eq_top_iff_one _).mp span_eq; clear span_eq
wlog finset_eq : ∃ t : Finset R, t = s generalizing s
· have ⟨t, hts, mem⟩ := Submodule.mem_span_finite_of_mem_span mem
have ⟨r, eq, uniq⟩ := this t (fun a ↦ f ⟨a, hts a.2⟩)
(fun a b ↦ h ⟨a, ht... |
12,779 | lean_file_chunk | 540d7e1dff6937497e8104c1d0c58639131146d78b12d99fb0148669aebeca19 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.NumberTheory.RamificationInertia.Basic
/-!
# Primes in an extension of localization at prime
Let `R ⊆ S` be an extension of Dedekind dom... |
23,929 | lean_file_chunk | 2dd1754b6bf0e853a0fc415f3b75037d3026c97bea4dfc0f393b114e6890c228 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Lattice
public import Mathlib.RingTheory.Ideal.Over
pub... |
4,598 | lean_file_chunk | 5bcdc7381de033fbdfe465b35773d3c16c17b953bcaaf267339db705938ac954 | lean_source_file | pkg_mathlib_source_file_chunks | simpa [ne_eq, Ideal.Quotient.eq_zero_iff_mem] using Ideal.mem_primeCompl_iff.mp s.prop
have h₂ : equivQuotMaximalIdeal p Rₚ (Ideal.Quotient.mk p ↑s) ≠ 0 := by
rwa [RingEquiv.map_ne_zero_iff]
rw [RingEquiv.symm_apply_eq, ← mul_left_inj' h₂, map_mul, mul_assoc, ← map_mul,
inv_mul_cancel₀ h₁, map_one, mul_one,... |
7,454 | lean_file_chunk | e9c77c2efd02da9d381598f83bf5921c0a021eeb04faf511bb23df83568ef517 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Kexing Ying
-/
module
public import Mathlib.Analysis.Normed.Order.UpperLower
public import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
public import... |
7,506 | lean_file_chunk | f3990844211d6ac9e19f76d513fa9abdc8bd0940b8fc59c7ea8ae86a7ab3fb13 | 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.Measure.AEMeasurable
/-!
# Typeclasses for measurability of lattice operations
In this file we define classes `MeasurableSup... |
9,852 | lean_file_chunk | 0d0aa0b47debafc22093e87373307ed8e078a70c716b4bf043df931404c2964d | 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, Mario Carneiro
-/
module
public import Mathlib.MeasureTheory.OuterMeasure.OfFunction
public import Mathlib.MeasureTheory.PiSystem
/-!
# The Carathéodory σ-algebra of ... |
5,350 | lean_file_chunk | e2061ea86d4ab326a7cd7090c75d9fdf942a4c5ffda81b090fc253616670ceda | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.OuterMeasure.AE
/-!
# Borel-Cantelli lemma, part 1
In this file we show one implication of the **Borel-Ca... |
21,364 | lean_file_chunk | 3a7eed5db1b16992b4485341c331b0a39c43f02f7627ea9087c8ecce490bd4a2 | 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, Mario Carneiro
-/
module
public import Mathlib.MeasureTheory.OuterMeasure.Operations
public import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from fun... |
8,605 | lean_file_chunk | 104b0412eb6bdd5ef744657974ff61d916998ebb9fe25a632537b764947705d4 | 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, Mario Carneiro
-/
module
public import Mathlib.Data.Countable.Basic
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Order.Disjointed
public import Mat... |
12,762 | lean_file_chunk | ab54e57e3e349e5260ee9b887b763f53a0d7f995947f0cc825a7dc435db4287b | 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, Mario Carneiro
-/
module
public import Mathlib.Algebra.Order.Group.Indicator
public import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# Operations on outer measures... |
9,476 | lean_file_chunk | 611389fd037017b6d48b8fa4209110879e0bee7d6f822102cc5af68f40a3c9f2 | 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, Peter Pfaffelhuber
-/
module
public import Mathlib.MeasureTheory.SetSemiring
public import Mathlib.MeasureTheory.Measure.AddContent
public import Mathlib.MeasureTheory.Mea... |
2,619 | lean_file_chunk | 0def42763218df7f0eb822784632c0891babd26599678b580b5f3d9eee225f8c | 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, Mario Carneiro, Yury Kudryashov
-/
module
public import Mathlib.Topology.Algebra.InfiniteSum.Defs
public import Mathlib.Topology.Order.Real
/-!
# Definitions of an ou... |
10,030 | lean_file_chunk | 9e0808d94a973e83fa2fa7040f7eb47b8fb633a6ec187f45b71c4739ff931231 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or ... |
18,772 | lean_file_chunk | 386757c6e72a55418ab64389380cc342875f5c750e6e67fb4249f85e9e15506b | 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, Mario Carneiro
-/
module
public import Mathlib.Data.ENNReal.Action
public import Mathlib.MeasureTheory.MeasurableSpace.Constructions
public import Mathlib.MeasureTheor... |
23,965 | lean_file_chunk | c215c3cc8a3705bf518368d471804bb25381881d90db545c6fc111db124157d4 | 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.Real
public import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
public import Mathlib.Topology.Algebra.InfiniteSum.M... |
23,989 | lean_file_chunk | f74a414c1df531544d8b5f4c8a7995c4024f8f45956deeefbb3c4979c0544e2a | lean_source_file | pkg_mathlib_source_file_chunks | def restrict (v : VectorMeasure α M) (i : Set α) : VectorMeasure α M :=
if hi : MeasurableSet i then
{ measureOf' := fun s => if MeasurableSet s then v (s ∩ i) else 0
empty' := by simp
not_measurable' := fun _ hi => if_neg hi
m_iUnion' := by
intro f hf₁ hf₂
convert v.m_iUnion (fu... |
2,698 | lean_file_chunk | 6f97b8ba0cbbc6241ca0d9da6752293c60af5b81efec3bbb7cabb5a59775095a | lean_source_file | pkg_mathlib_source_file_chunks | theorem toMeasureOfLEZero_real_apply (hi : s ≤[i] 0) (hi₁ : MeasurableSet i)
(hj₁ : MeasurableSet j) :
(s.toMeasureOfLEZero i hi₁ hi).real j = -s (i ∩ j) := by
simp [measureReal_def, toMeasureOfLEZero_apply _ hi hi₁ hj₁]
/-- `SignedMeasure.toMeasureOfZeroLE` is a finite measure. -/
instance toMeasureOfZeroLE... |
16,348 | lean_file_chunk | d53d6387e959d46c8c7b0c720d64cbb532075071187fa4baacd0402ee3d6a35a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Group.InfiniteSum
public import Mathlib.MeasureTheory.Measure.AddContent
public import Mathlib.MeasureTheory.Measu... |
9,162 | lean_file_chunk | 8c59b6665592fab0dd2cab759ed56b3c83756064235811b890b47a94a8415afb | 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.MeasureTheory.Function.AEEqOfIntegral
/-!
# Vector measure defined by an integral
G... |
13,963 | lean_file_chunk | 8aaa29ceec5511def4aadd3d7121b8b44aa5f1091247a678b790d313640281a1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Group.Defs
public import Mathlib.MeasureTheory.Measure.Stieltjes
public import Mathlib.MeasureTheory.VectorMeasure... |
2,455 | lean_file_chunk | 3ab4e31da83d6f9716c039f9631f7497953784a926549cc9b43854e867dd28a1 | 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.Integral.Bochner.ContinuousLinearMap
/-!
# Integrability in a product space
We prove that `f : X → Π i, E i` is in `Lᵖ` ... |
6,750 | lean_file_chunk | 8a01bb61f74122d58c60631dc66a036ccb8f5865fb3f1df46dfe4ec1b682bc76 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
module
public import Mathlib.Topology.ContinuousMap.Compact
public import Mathlib.Topology.ContinuousMap.Algebra
public import Mathlib.MeasureTheory.Integral.Int... |
6,214 | lean_file_chunk | e40ed96ba8a467d71cd8ff5240a71c5b0541c4352397afcf0a1810233813772c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
module
public import Mathlib.Topology.ContinuousMap.ContinuousMapZero
public import Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap
/-!
# Specific results... |
3,547 | lean_file_chunk | 655814e2a82fb9ec5a4d0a12d5f8a0c08e0df5089a34d4cc5a6592517087dd13 | 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.Analysis.Normed.Lp.PiLp
public import Mathlib.MeasureTheory.SpecificCodomains.Pi
/-!
# Integrability in `WithLp`
We prove that `f : X ... |
2,687 | lean_file_chunk | fc09855a07a98208a2ed736400e54f20fbeb63b9278251e3d1a6702a1bd25e32 | 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, Mario Carneiro
-/
module
public import Mathlib.MeasureTheory.MeasurableSpace.Defs
public import Mathlib.GroupTheory.GroupAction.IterateAct
public import Mathlib.Data.R... |
10,026 | lean_file_chunk | 57a67327c8f10e4e785968da55f3bc94bfd453f35c30ed8e68e444b8073c68b5 | 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, Violeta Hernández Palacios
-/
module
public import Mathlib.MeasureTheory.MeasurableSpace.Defs
public import Mathlib.SetTheory.Cardinal.Regular
public import Math... |
12,157 | lean_file_chunk | b22b3cb38024f6287120775b2ce131362b55753f0ebab0f95004b6f4fde6d4a9 | 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, Mario Carneiro
-/
module
public import Mathlib.MeasureTheory.MeasurableSpace.Constructions
public import Mathlib.Order.Filter.AtTopBot.CompleteLattice
public import Ma... |
23,971 | lean_file_chunk | 97d1f26d07bf056be6a9387a2c2fd01e936df4ad18b100a1799233229cc670a4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher, Yury Kudryashov, Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.MeasurableSpace.Embedding
public import Mathlib.Data.Set.MemPartition
public import Mathl... |
1,741 | lean_file_chunk | b52faf0020212ba0e5ed0403d6e96c9ace4462f9cd6d9982261a0bc24240b188 | lean_source_file | pkg_mathlib_source_file_chunks | /-- A class registering that either `α` is countable or `β` is a countably generated
measurable space. -/
class CountableOrCountablyGenerated (α β : Type*) [MeasurableSpace β] :
Prop where
countableOrCountablyGenerated : Countable α ∨ MeasurableSpace.CountablyGenerated β
instance instCountableOrCountablyGenerate... |
5,423 | lean_file_chunk | 090c864b01c018de87a52e19402156c6008e9069aa042491f4925b2ae2439a20 | 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.MeasurableSpace.Embedding
public import Mathlib.MeasureTheory.PiSystem
/-!
# The product sigma algebra
This file tal... |
850 | lean_file_chunk | 1495b05aabc743ad923fbad2557e57533b943b97062f3004d69e46f3bb0d9699 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Data.Set.Card
public import Mathlib.MeasureTheory.MeasurableSpace.Constructions
/-!
# Measurability of `Set.encard` and `Set.ncard`
... |
4,956 | lean_file_chunk | 41514165bdb424cad6abf67ad425c80666e33a5e8063a607bc38c2bfbf9dad1b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
module
public import Mathlib.MeasureTheory.MeasurableSpace.Defs
public import Mathlib.Order.Filter.CountableInter
/-!
# Measurability modulo a filter
In this fil... |
4,633 | lean_file_chunk | b52a941b818a2445d338f1ba791c8ed49791c0fef5231680ce998eef4f128c0a | 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.MeasurableSpace.Constructions
public import Mathlib.MeasureTheory.PiSystem
/-!
# Bases of the indexed product σ-algeb... |
14,202 | lean_file_chunk | b3a54d7dd195c97703c8680bdacb7c1bb6e53e8cfdcaaaaa2e402074e95bb461 | 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, Mario Carneiro
-/
module
public import Mathlib.Algebra.Notation.Indicator
public import Mathlib.Data.Int.Cast.Pi
public import Mathlib.Data.Nat.Cast.Basic
public impor... |
23,958 | lean_file_chunk | 5959de69304f84013ebff148a62d9baaba94ee230223ae82e2ea078b3865d971 | 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, Mario Carneiro
-/
module
public import Mathlib.Data.Finset.Update
public import Mathlib.Data.Prod.TProd
public import Mathlib.Data.Set.UnionLift
public import Mathlib.... |
18,026 | lean_file_chunk | 287865c398cafc6b28dd87e58ebb2da3edbc7804a84fe7922be36098637980eb | lean_source_file | pkg_mathlib_source_file_chunks | instance MeasurableSpace.pi [m : ∀ a, MeasurableSpace (X a)] : MeasurableSpace (∀ a, X a) :=
⨆ a, (m a).comap fun b => b a
variable [∀ a, MeasurableSpace (X a)] [MeasurableSpace γ]
theorem measurable_pi_iff {g : α → ∀ a, X a} : Measurable g ↔ ∀ a, Measurable fun x => g x a := by
simp_rw [measurable_iff_comap_le, ... |
22,605 | lean_file_chunk | 6975cf1b2cfaaab1e6524d5ecf1250a284196290052fd10b3d5eb048f46e340c | 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, Mario Carneiro
-/
module
public import Mathlib.Data.Set.Countable
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
public import Mathlib.Tactic.FunProp.A... |
23,936 | lean_file_chunk | b86240549fed655b3cdd8dd684f6579bd6fd717173af3a6363d05c8d4180b3f9 | 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, Mario Carneiro
-/
module
public import Mathlib.MeasureTheory.MeasurableSpace.Constructions
public import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equiva... |
11,093 | lean_file_chunk | 09eb44469e5c327ec0a92798204797b97669f5ba4523378b85ec418cc821bfe1 | lean_source_file | pkg_mathlib_source_file_chunks | `{i // p i}` and `{i // ¬p i}`. See also `Equiv.piEquivPiSubtypeProd`. -/
@[simps! -fullyApplied]
def piEquivPiSubtypeProd (p : δ' → Prop) [DecidablePred p] :
(∀ i, π i) ≃ᵐ (∀ i : Subtype p, π i) × ∀ i : { i // ¬p i }, π i where
toEquiv := .piEquivPiSubtypeProd p π
/-- The measurable equivalence between the pi t... |
1,174 | lean_file_chunk | d831718c6b79eb6ea317fa748726356fffb0f621e4bf7b5cc45b8598639e1458 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.MeasurableSpace.Constructions
public import Mathlib.Order.Restriction
/-!
# Measurability of the restriction function for... |
2,800 | lean_file_chunk | deb5cfb927702bf4e46dc9671738053f8a587e394187ed128b4482d23b7bd652 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.MeasurableSpace.Defs
/-!
# σ-algebra of sets invariant under a self-map
In this file we define `MeasurableSpace.invaria... |
4,524 | lean_file_chunk | 211226983d91bb517c86b3f09d11b48ce49f3dfd6b13aa6f8e9c464c18a27bc2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.MeasureTheory.Measure.GiryMonad
public import Mathlib.CategoryTheory.Monad.Algebra
public import Mathlib.Topology.Category.TopCat.Basic
... |
8,164 | lean_file_chunk | 717d02fad245921556479e500fed236a4858e91af6f02f086493d814d728b13c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
module
public import Mathlib.MeasureTheory.Group.Defs
public import Mathlib.MeasureTheory.Measure.Prod
/-!
# The multiplicative and additive convolution of measures
In... |
7,073 | lean_file_chunk | c06b5a0bfb396c9a584d29a200a78e220c7bdb726a6c051efc593478d0fae126 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Integral.Bochner.Basic
public import Mathlib.MeasureTheory.Group.Measure
/-!
# Bochner Integration on Groups
We deve... |
3,180 | lean_file_chunk | 70203285cad636775f6dbe1c84ad09b86cb4a1f0e45295e82b0ee6e4c6e08696 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Group.Action
public import Mathlib.Order.Filter.EventuallyConst
/-!
# A.e. stabilizer of a set
In this file we define ... |
5,094 | lean_file_chunk | 3f75d79779466f3f28ecf454ecc18dca8b09080323d3afb4a2fdd3fe1f3700f6 | 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.Integral.IntervalIntegral.Periodic
public import Mathlib.Data.ZMod.QuotientGroup
public import Mathlib.MeasureTheory.Group.AESta... |
23,980 | lean_file_chunk | 52818516127461daff727c67d1969f1a5302eb9715866f524ed5dcf5151f9451 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.Group.Measure
public import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this ... |
1,528 | lean_file_chunk | a1243633792fb83793712322cabd8555568afa13f9d0abd4cfe4ca46e836821d | 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, Alex J. Best
-/
module
public import Mathlib.MeasureTheory.Group.Arithmetic
/-!
# Pointwise set operations on `MeasurableSet`s
In this file we prove several versio... |
6,723 | lean_file_chunk | b5cc349b3c906b6534036a71a30a86170d88c5a61116893cab4dd8823b7a5904 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
module
public import Mathlib.MeasureTheory.Function.LocallyIntegrable
public import Mathlib.MeasureTheory.Group.Integral
public import Mathlib.MeasureTheory.Group.Measure
publ... |
1,787 | lean_file_chunk | 09fc331df36a27d3b9c106f9060a7b5fe301120a42071dfb64975a83a22fe023 | 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.Group.Convolution
public import Mathlib.MeasureTheory.Integral.Prod
/-!
# Bochner integrals of convolutions
This file contai... |
7,300 | lean_file_chunk | 06081a1f74f213cbb3478f1bca53c66dd8db088910285a626fb9a914f5855ec7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best
-/
module
public import Mathlib.Analysis.Convex.Body
public import Mathlib.Analysis.Convex.Measure
public import Mathlib.MeasureTheory.Group.FundamentalDomain
/-!
# Geomet... |
23,997 | lean_file_chunk | ad74b2366ebba2f88f352d4181b903f9d824d7cefe5a8f0bfdbeb83dfcc791fc | 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.MeasureTheory.Measure.AEMeasurable
/-!
# Typeclasses for measurability of operations
In this file we define classes `MeasurableMul` ... |
12,559 | lean_file_chunk | e31fa2cca230c055b8358ae0f13f84da8d1478c02a0baaabfb79d8e8d8d36cc8 | lean_source_file | pkg_mathlib_source_file_chunks | instance (priority := 100) MeasurableSMul₂.toMeasurableSMul [MeasurableSMul₂ M X] :
MeasurableSMul M X where
variable [MeasurableSMul M X]
@[to_additive (attr := fun_prop)]
theorem Measurable.smul_const (hf : Measurable f) (y : X) : Measurable fun x => f x • y :=
(MeasurableSMul.measurable_smul_const y).comp hf... |
11,452 | lean_file_chunk | 7826f7f140585c2bbe0c0e96729049de657ef611aed82e50e59fef465d653c84 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Stefano Rocca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Stefano Rocca
-/
module
public import Mathlib.MeasureTheory.Group.Defs
public import Mathlib.MeasureTheory.Group.Action
public import Mathlib.MeasureTheory.Measure.Typecl... |
4,604 | lean_file_chunk | b5611344678b9a49f607edf5f008d589e3ff2001ede1e7914fd99c490929b271 | 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, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Measure.Map
/-!
# Definitions about invariant measures
In this file we define typeclasses for measu... |
3,419 | lean_file_chunk | 3b9fbd8f3f6e3bd807f928e01553536a097cf587d513b8a4ad1a24c241232269 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Group.Measure
/-!
# Lebesgue Integration on Groups
We develop properties of integrals with a group as domain.
This f... |
8,945 | lean_file_chunk | b3f8394f961d30a94ba91901c2f09d77613c83cbd82b39e401333cf285e9b691 | 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.MeasureTheory.Group.Arithmetic
/-!
# (Scalar) multiplication and (vector) addition as measurable equivalences
In this file we define... |
14,042 | lean_file_chunk | d8d135006d00afd3ef1a38172ffcb9226381ceee5df866df6aa2ea0ec189b296 | 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.Dynamics.Ergodic.MeasurePreserving
public import Mathlib.Dynamics.Minimal
public import Mathlib.GroupTheory.GroupAction.Hom
public imp... |
23,988 | lean_file_chunk | a72630e67dd57ef002e82bca468b7cddf3b95835c57b3349ca703b15ff325037 | 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.Algebra.Group.Pointwise.Set.Card
public import Mathlib.GroupTheory.Complement
public import Mathlib.MeasureTheory.Group.Action
publi... |
18,242 | lean_file_chunk | 90bc6d7ffda179d4b7aa9b549a3c2614e44c9844f2cabacfb164f6a249b6e8a5 | lean_source_file | pkg_mathlib_source_file_chunks | IsOpenPosMeasure μ :=
let ⟨K, hK, h2K⟩ := Regular.exists_isCompact_not_null.mpr (NeZero.ne μ)
isOpenPosMeasure_of_mulLeftInvariant_of_compact K hK h2K
/-- A nonzero left-invariant inner regular measure gives positive mass to any open set. -/
@[to_additive
/-- A nonzero left-invariant inner regular measure gives po... |
23,961 | lean_file_chunk | d50d473927079fa0d318b1ef1db52ea3313aca597076196ea569d3caee3a5f99 | 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, Alex Kontorovich, Heather Macbeth
-/
module
public import Mathlib.MeasureTheory.Group.Action
public import Mathlib.MeasureTheory.Group.Pointwise
public import Mathli... |
19,819 | lean_file_chunk | 974a4cdf5d635691f82e94d35e64037941be5fbb88abf7852d5c3f81d9deb9d9 | lean_source_file | pkg_mathlib_source_file_chunks | (hf : ∀ γ : G, ∀ x : α, f (γ • x) = f x) : essSup f (μ.restrict s) = essSup f μ := by
refine le_antisymm (essSup_mono_measure' Measure.restrict_le_self) ?_
rw [essSup_eq_sInf (μ.restrict s) f, essSup_eq_sInf μ f]
refine sInf_le_sInf ?_
rintro a (ha : (μ.restrict s) {x : α | a < f x} = 0)
rw [Measure.restrict_... |
4,245 | lean_file_chunk | fb2fd258a25bcf87026a789927d326f1a54f560bf8aadfd4f24b3c83a16c4929 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Gaëtan Serré
-/
module
public import Mathlib.MeasureTheory.Measure.Haar.Unique
/-!
# The canonical measure on the unit interval
This file provides a `MeasureTheory.Measure... |
3,698 | lean_file_chunk | 470c200c7d0c4e139954c3fde067a5a49260a145d2e5b776c1d00cea75705d4c | 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, Peter Pfaffelhuber
-/
module
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
public import Mathlib.MeasureTheory.Constructions.Cylinders
/-! # Cylinder... |
1,841 | lean_file_chunk | 201eb4b1f423b5d6cad1265e63dd30e9e13abcaa1e0876e20ca06d4f6b307b8a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Basic
public import Mathlib.MeasureTheory.MeasurableSpace.Embedding
/-!
# Sigma-algebra on simple graphs
In this... |
23,996 | lean_file_chunk | 86fa88cb03a9d8bf54ba1c1d59002c27d29336684a41cf39b58ac6c67df51806 | 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.Algebra.BigOperators.Fin
public import Mathlib.Logic.Encodable.Pi
public import Mathlib.MeasureTheory.Group.Measure
public import Ma... |
21,216 | lean_file_chunk | 5787b6b8c9b7500995035b577f43a20cba2c075c43bd4b458491e3c6186c10d7 | lean_source_file | pkg_mathlib_source_file_chunks | refine ⟨fun x => ?_⟩
choose s hxs ho hμ using fun i => (μ i).exists_isOpen_measure_lt_top (x i)
refine ⟨pi univ s, set_pi_mem_nhds finite_univ fun i _ => IsOpen.mem_nhds (ho i) (hxs i), ?_⟩
rw [pi_pi]
exact ENNReal.prod_lt_top fun i _ => hμ i
instance {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, Measure... |
13,597 | lean_file_chunk | 7d2a3b5589a18ee7ff7fd336e50d8b3ac24d8510c7fb1939c7c2e5b2d37aeaf1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Algebra.Order.Field.Pointwise
public import Mathlib.Analysis.Normed.Module.Ball.RadialEquiv
public import Mathlib.Analysis.SpecialFunc... |
7,857 | lean_file_chunk | 24785cc84b41e87254d9da6b46c902c6dd8c9f73eeed48ee30289545b55c7737 | 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, Peter Pfaffelhuber
-/
module
public import Mathlib.MeasureTheory.Constructions.Projective
public import Mathlib.MeasureTheory.Measure.AddContent
public import Mathlib.Meas... |
723 | lean_file_chunk | 961625fed106112f3c6825655144c013dbc4ca950866e2ad19283f92578286c8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.LinearAlgebra.Quotient.Defs
public import Mathlib.MeasureTheory.MeasurableSpace.Constructions
/-!
# Measurability on the quotient of a modu... |
6,319 | lean_file_chunk | 86202b984ddcdba7f58fa1d47ccb8e12609c3e3f894183551aea5ab9200587cb | 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, Peter Pfaffelhuber
-/
module
public import Mathlib.MeasureTheory.Constructions.Cylinders
public import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
/-!
# Project... |
20,999 | lean_file_chunk | 7ae8bddd8937de50152253528eefe4535949157cb0bc7edc580cd3bd4a096eb7 | 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, Peter Pfaffelhuber, Yaël Dillies, Kin Yau James Wong
-/
module
public import Mathlib.MeasureTheory.MeasurableSpace.Constructions
public import Mathlib.MeasureTheory.PiSyst... |
932 | lean_file_chunk | 10ee5ba4e12219a6c5539e3cc9b6cad35681ba411fc5744facce0e5e4acda7e9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Group.AddChar
public import Mathlib.MeasureTheory.MeasurableSpace.Defs
/-!
# Measurable space instance for additive characters
Thi... |
23,954 | lean_file_chunk | 441f7ae68a1e78f8b9ed9ca178c0de7ddbc77d41f4c26c77c762e656c9090f2f | 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.Analysis.Calculus.FDeriv.Congr
public import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
public import Math... |
23,919 | lean_file_chunk | e56f44955a802d5df2b7c1c894325fedc20ef615107319e4dd44ebb7071c458f | lean_source_file | pkg_mathlib_source_file_chunks | simpa only [tsub_zero, inv_eq_zero, Ne] using H
simp only [mul_zero] at this
exact (tendsto_order.1 this).2 δ₀ δ₀pos
-- let `δ` be small enough, and `f` approximated by `B` up to `δ`.
filter_upwards [L1, L2]
intro δ h1δ h2δ s f hf
have hf' : ApproximatesLinearOn f (B : E →L[ℝ] E) s δ := by convert hf
... |
19,420 | lean_file_chunk | fba1038c8d7f96c637d9cc06a690e739ec1fc5027ab03ab387df78ac7ad5eb41 | lean_source_file | pkg_mathlib_source_file_chunks | `spanningSets μ`, and apply the previous result to each of these parts. -/
let u n := disjointed (spanningSets μ) n
have u_meas : ∀ n, MeasurableSet (u n) := by
intro n
apply MeasurableSet.disjointed fun i => ?_
exact measurableSet_spanningSets μ i
have A : s = ⋃ n, s ∩ u n := by
rw [← inter_iUnio... |
23,993 | lean_file_chunk | 57ffa832f039d897b724c619798248a9a8f40ce8f70f93f83325a9a1c9dc00df | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
module
public import Mathlib.Dynamics.Ergodic.MeasurePreserving
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStron... |
13,486 | lean_file_chunk | 32fbb6c33dfce7b8d5f6573ccc7042291b0a06e7af2118352d144f2eedeb1d3b | lean_source_file | pkg_mathlib_source_file_chunks | filter_upwards [coeFn_sup f g] with _ ha
rw [ha]
exact le_sup_right
protected theorem sup_le (f g f' : α →ₘ[μ] β) (hf : f ≤ f') (hg : g ≤ f') : f ⊔ g ≤ f' := by
rw [← coeFn_le] at hf hg ⊢
filter_upwards [hf, hg, coeFn_sup f g] with _ haf hag ha_sup
rw [ha_sup]
exact sup_le haf hag
end Sup
section Inf
va... |
5,370 | lean_file_chunk | 4af84c2bf12ec415e3aa630fb3aaf924ac7124c75697f8e60b9c693353a2f4ad | 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.StronglyMeasurable
public import Mathlib.Probability.Process.Filtration
/-!
# Factorization of a map... |
18,462 | lean_file_chunk | 330fc8fd5a64a1d310f64c7f2982b4e86165706f019697b7083a27cb0da11bfa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Igor Khavkine. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Igor Khavkine
-/
module
public import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
public import Mathlib.MeasureTheory.Function.UniformIntegrable
/-!
# Uniform tightness
This ... |
23,984 | lean_file_chunk | 7f3f3d21752ae73c502cbc9fa87146353ed045fc22f8cb2b53a393ee62cdfc12 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Algebra.Algebra.Pi
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A functio... |
23,963 | lean_file_chunk | a79fed4c8e40174a890b9dba4965560b5b820eaaca68f8b09cccf1709af595b3 | lean_source_file | pkg_mathlib_source_file_chunks | instance [InvolutiveStar β] : InvolutiveStar (α →ₛ β) where
star_involutive _ := ext fun _ ↦ star_star _
instance [AddMonoid β] [StarAddMonoid β] : StarAddMonoid (α →ₛ β) where
star_add _ _ := ext fun _ ↦ star_add ..
instance [Mul β] [StarMul β] : StarMul (α →ₛ β) where
star_mul _ _ := ext fun _ ↦ star_mul ..
... |
10,130 | lean_file_chunk | 9e8180bc97ea19d8c97877d5e18ecc9ac284f92eed1fa29c274ce6d971633b14 | lean_source_file | pkg_mathlib_source_file_chunks | refine finMeasSupp_iff.2 fun b hb => ?_
rw [f.lintegral_eq_of_subset' (Finset.subset_insert b _)] at h
refine ENNReal.lt_top_of_mul_ne_top_right ?_ hb
exact (lt_top_of_sum_ne_top h (Finset.mem_insert_self _ _)).ne
theorem iff_lintegral_lt_top {f : α →ₛ ℝ≥0∞} (hf : ∀ᵐ a ∂μ, f a ≠ ∞) :
f.FinMeasSupp μ ↔ f.lint... |
4,195 | lean_file_chunk | 61c1524ec124fa9916265611055f93259105e790df32a5fe4ed2dff1cbab8c40 | 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
-/
module
public import Mathlib.Data.Setoid.Partition
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
/-!
# Measurability of piecewise fun... |
3,856 | lean_file_chunk | 91de38ca4b539c3b780113c62dd1436bb9d601a2e231e64439c5c75c3bbf2844 | 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.Normed.Order.Lattice
public import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
public import Mathlib.MeasureTheory.Function... |
6,959 | lean_file_chunk | 8f851d808db9b04628d380d74f74c742002efdbabd33060131553ae361acdf26 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.MeasureTheory.Integral.Average
/-!
# Bergelson's intersectivity lemma
This file proves the Bergelson intersectivity lemma: In a finite mea... |
23,983 | lean_file_chunk | f7aeb9793a763822a361fc78b543ab2fd1fb69aa52844756707260b409a6bb98 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Function.ConvergenceInMeasure
public import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This... |
23,979 | lean_file_chunk | 212532be112cfe71718e22f15876818ba952a4e8f2a0bcc5783eab1f11004afa | lean_source_file | pkg_mathlib_source_file_chunks | exact measure_mono (Set.subset_univ _)
· exact Real.rpow_pos_of_pos (measureUnivNNReal_pos hμ) _
· positivity
have : ENNReal.ofReal (ε.toReal / 3) = ε / 3 := by
rw [ENNReal.ofReal_div_of_pos (show (0 : ℝ) < 3 by simp), ENNReal.ofReal_toReal h.ne]
simp
rw [this] at hnf hng hlt
rw [eLpNorm_neg, ← ... |
1,102 | lean_file_chunk | ce71bff6d0867aee33cae720f9b17154268eb71c2158b132892719c579d0243c | lean_source_file | pkg_mathlib_source_file_chunks | exact ⟨C, fun ⟨f, s, hs⟩ => Lp.eLpNorm_le_of_ae_tendsto
(Eventually.of_forall fun n => hC (s n)) (fun n => hUI.1 (s n)) hs⟩
/-- Suppose `f` is a sequence of functions that converges a.e. to `g`. If `f` is
`UniformIntegrable`, then `g` is in `Lp`. -/
lemma UniformIntegrable.memLp_of_ae_tendsto {κ : Type*} {u : Fi... |
19,094 | lean_file_chunk | 6b689491140f07b359aa89bb3ffd9430d086fd7caec6f95c5644297d0e3917a2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
module
public import Mathlib.MeasureTheory.Measure.Regular
public import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
public import Mathlib.Topology.UrysohnsLe... |
10,067 | lean_file_chunk | ed03c124caab031d4745679e964278055ebc00b8fb7d16aa7f9a46535bb0d8f5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.MeasureTheory.Integral.Bochner.Basic
/-! # Continuous bilinear maps on `MeasureTheory.Lp` spaces
Given a continuous bilinear map `B : E ... |
11,728 | lean_file_chunk | 26c70ed251e8e0ca68e60a37826b8b84ac1d719e9fc4aba4f2fc30cd1e78d27e | 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.StronglyMeasurable.ENNReal
public import Mathlib.MeasureTheory.Measure.WithDensity
/-! # From equality of integrals ... |
13,322 | lean_file_chunk | 3940348a73bdc212da4c01c24248ca6dddf93b03e37a7ee15fcda491b075498f | 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.Constructions.BorelSpace.Order
public import Mathlib.MeasureTheory.Measure.Count
public import Mathlib.Order.Filter.ENNReal
pu... |
23,731 | lean_file_chunk | abd74f06a6704ab702b6fc551bea55b654ea78ff98cdc6fea158ba84b684d4d5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. 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.Egorov
public import Mathlib.MeasureTheory.Function.LpSpace.Complete
/-!
# Convergence in ... |
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