chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
13,389 | lean_file_chunk | fac10b69b047daf0bfdca2b2c5ac2fd6907984ebbf415ab438f11a3372bf5666 | lean_source_file | pkg_mathlib_source_file_chunks | compMeasurePreserving f^[n] (MeasurePreserving.iterate hf n) := by
funext
induction n with
| zero => simp
| succ n h =>
nth_rewrite 1 [add_comm n 1]
simp [Function.iterate_add, h, compMeasurePreserving_comp (hf.iterate n) hf]
variable (𝕜 : Type*) [NormedRing 𝕜] [Module 𝕜 E] [IsBoundedSMul 𝕜 E]
/--... |
4,701 | lean_file_chunk | 0e22261bf11290944df42fa548378cb9911145ed6ae2d7dd34a5e14a070c3569 | 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.Function.SimpleFuncDenseLp
public import Mathlib.MeasureTheory.Measure.ContinuousPreimage
/-!
# Continuity of `MeasureT... |
8,665 | lean_file_chunk | 81da935dc32b31f70d25b5d1c5a0944f42f6ddaa812a99bb68b2befbfcb191d9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Operator.NormedSpace
public import Mathlib.MeasureTheory.Function.LpSpace.Basic
public import Mathlib.Mea... |
3,337 | lean_file_chunk | af8d3251147aab3203ec876ff9ef0f5a52c57789a65e9980e8753096f072f80c | 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.Function.AEEqFun
public import Mathlib.MeasureTheory.Group.Action
public import Mathlib.GroupTheory.GroupAction.DomAct.B... |
2,573 | lean_file_chunk | 74c612fcc192526e737f6c59e589378987aa03e44a5e15ce1ce1900113190d36 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
/-!
# Lp seminorm with respect to trimmed measure
In this file we prove basic properties of the Lp... |
4,364 | lean_file_chunk | 3ecf6dbf99ed5911185954fff8062cb0801624493e6e59238b8944100538a039 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity
/-!
# Scalar multiplication on ℒp space
-/
public noncomputable section
... |
2,053 | lean_file_chunk | d205e3927e91287f238b01d878aa34270097e7e7533a5a4804a0d7ff093afb6b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.MeasureTheory.Function.LpSeminorm.Indicator
/-!
# `L^p`-seminorms on `count` and `dirac`
-/
@[expose] public section
open Measure... |
10,131 | lean_file_chunk | 1a4e9439343d715c9c6759f5cc309ae3d18d7958c38a175f523c79d0f0e35159 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
public import Mathlib.Analysis.Normed.Group.Indicator
public import Mathlib.Measu... |
1,233 | lean_file_chunk | 5fd38c419f2914f4425c08e63b9b93cd45b0c4a6ddc93d619485378d208473ba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
public import Mathlib.MeasureTheory.Measure.Prod
/-!
# ℒp spaces and products
-/
public section
... |
11,533 | lean_file_chunk | fb0af1d9b62e74dc8e1ece58466bf18180d29880cef8a8173d29c1fd04a73386 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
/-!
# Monotonicity and ℒp seminorms
-/
public noncomputable section
open Topol... |
23,923 | lean_file_chunk | 9918662c6026f8fba1bcf5362a5a2478ef0c9ed03a3f18e832ec98b2fe699317 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.Data.Fintype.Order
public import Mathlib.MeasureTheory.Function.AEEqFun
public import Mathlib.MeasureTheory.Function.LpSe... |
18,118 | lean_file_chunk | dbabed4e056e215b75c5ca5e1f8d445b0cc6b0f8562ea124930dc49ea8ccc5a0 | lean_source_file | pkg_mathlib_source_file_chunks | (memLp_const C).of_le hf (hfC.mono fun _x hx => le_trans hx (le_abs_self _))
theorem memLp_of_bounded [IsFiniteMeasure μ]
{a b : ℝ} {f : α → ℝ} (h : ∀ᵐ x ∂μ, f x ∈ Set.Icc a b)
(hX : AEStronglyMeasurable f μ) (p : ENNReal) : MemLp f p μ :=
have ha : ∀ᵐ x ∂μ, a ≤ f x := h.mono fun ω h => h.1
have hb : ∀ᵐ x ... |
4,065 | lean_file_chunk | f3f1fe79eac30d1266187be7b33e72534c122c7e2cfe46ed4c742eb276669455 | 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.LpSeminorm.Basic
/-!
# Chebyshev-Markov inequality in terms of Lp seminorms
In this file we formulate several version... |
6,379 | lean_file_chunk | 2ab9f9c0595e34a2ec18695988f3d9447030fe12cfaab669f3a90ff11ba6dcd8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
public import Mathlib.MeasureTheory.Function.EssSup
public import Mathlib.MeasureThe... |
16,417 | lean_file_chunk | f8195bb4c8cfc6901943af9bbdb00159f61241d1f347f11ddded829a36f18eb9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Eric Wieser
-/
module
public import Mathlib.Data.ENNReal.Holder
public import Mathlib.MeasureTheory.Function.LpSeminorm.Indicator
public import Mathlib.MeasureTheory.Funct... |
7,115 | lean_file_chunk | a6bb5622a548a310a72ea5390d94ba087cbfde47cdabd7aec95ffe6471b3a95f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
public import Mathlib.MeasureTheory.Integral.MeanInequalities
/-!
# Triangle inequality for `Lp`-se... |
11,956 | lean_file_chunk | 2d32137f865650f94051a216c64f5bd591be1fe863e4fb2ff378dace996c18d7 | 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.MeasureTheory.Integral.Bochner.Basic
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Tactic.Positivity.Finset
/-!
# Real-valued Lᵖ no... |
4,490 | lean_file_chunk | 2e2f3aa86f57f1a01061d5708d5eb71fec48998ec4c4b7082d61e58960d2c21c | 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.Function.AEEqFun.DomAct
public import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-!
# Action of `Mᵈᵐᵃ` on `Lᵖ` s... |
2,094 | lean_file_chunk | 03a8450455e2a7bfce14e4f592bfc2a34b10bbee9ef1a927f9e502660438d936 | 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.Function.LpSpace.DomAct.Basic
public import Mathlib.MeasureTheory.Function.LpSpace.ContinuousCompMeasurePreserving
publi... |
5,858 | lean_file_chunk | 199e221cecb2971a6b4e1bdb988600555f5ece4a48e57a0e8cd587b879eea261 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
module
public import Mathlib.MeasureTheory.Measure.Haar.Unique
/-!
# Scaling Haar measure by a continuous isomorphism
If `G` is a locally compact topological group a... |
13,781 | lean_file_chunk | ca31fe491a0c7971e3a095abfd719258a79f52c0ebf71b3b9640dda914163b73 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.Analysis.InnerProductSpace.Basic
public import Mathlib.MeasureTheory.Group.Integral
public import Mathlib.MeasureTheory.Integral.Riesz... |
23,991 | lean_file_chunk | c45a2705de71de2dc852eade461a9b0eb92a643b085d3783bdbf86750fc3bdfa | 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.Measure.Content
public import Mathlib.MeasureTheory.Group.Prod
public import Mathlib.Topology.Algebra.Group.Compact
/... |
10,211 | lean_file_chunk | d988c96e53fe18f70d7a55dadcfa5390a5a85d4badbf544d5683d04f8753c709 | lean_source_file | pkg_mathlib_source_file_chunks | (haarContent_outerMeasure_self_pos K₀).trans_le (OuterMeasure.mono _ subset_closure)
end haar
open haar
/-!
### The Haar measure
-/
variable [TopologicalSpace G] [IsTopologicalGroup G] [MeasurableSpace G] [BorelSpace G]
/-- The Haar measure on the locally compact group `G`, scaled so that `haarMeasure K₀ K₀ = 1`. ... |
23,610 | lean_file_chunk | e103bfba3a5c2150dedb84f6ae15b6adac6fc9209e612b12454cbf867a8b1940 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Alex Kontorovich and Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
module
public import Mathlib.Algebra.Group.Opposite
public import Mathlib.MeasureTheory.Constructions.Polish.Basic
public im... |
23,989 | lean_file_chunk | 8f30b4ad809cfc1bd0bc9603df5ca3675069d1cb73f9e3b4ceae4559472b51cd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Function.LocallyIntegrable
public import Mathlib.MeasureTheory.Group.Integral
public import Mathlib.MeasureTheory.In... |
23,985 | lean_file_chunk | d4dceb33531657dd9df3878751f867a66175fefad1c2d6f3995e4c797c331679 | lean_source_file | pkg_mathlib_source_file_chunks | See `measure_isMulInvariant_eq_smul_of_isCompact_closure_of_measurableSet`.
Finally, if `s` has compact closure but is not measurable, its measure is the infimum of the
measures of its measurable supersets, and even of those contained in `closure s`. As `μ`
and `μ'` coincide on these supersets, this yields `μ s = μ' s... |
8,743 | lean_file_chunk | 340fe56b08b9a04061574892a0c7f492c52b7edcc1f667e0f9bec63dd4a094ef | lean_source_file | pkg_mathlib_source_file_chunks | @[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular [LocallyCompactSpace G]
(μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
[Regular μ] [Regular μ'] :
μ' = haarScalarFactor μ' μ • μ := by
have A : ∀ U, IsOpen U → μ' U... |
11,390 | lean_file_chunk | 9c4b5fd89ae8ec5e9002656cf2c673cda9626cef430fb597659171f673ec7705 | 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, Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
public import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
public i... |
13,093 | lean_file_chunk | e636d894a2f2439cf49109008e2c722756412f95c455bbdecbbd69d97b503f8c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Measure.Haar.Basic
public import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Additive Haar measure constructed fr... |
4,057 | lean_file_chunk | b490f4bddf4aba94611b41815e92b36c9e3fe4b40aeec98db8210e96201d1b13 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang, Yaël Dillies, Javier López-Contreras. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yaël Dillies, Javier López-Contreras
-/
module
public import Mathlib.MeasureTheory.Measure.Haar.Unique
/-!
# The distributive characte... |
8,120 | lean_file_chunk | 12ced1073310753a0091fba6bd20a1b85feb0ad61395b77e6b9fff2b7738b8f0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Measure.Haar.Basic
public import Mathlib.Analysis.Normed.Module.FiniteDimension
public import Mathlib.MeasureTheory.... |
11,732 | lean_file_chunk | 655572a755f8fb6d52433cedc6f9430109bd8ab88083bc0b7987cd790b2c4715 | 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.InnerProductSpace.Orientation
public import Mathlib.Analysis.InnerProductSpace.ProdL2
public import Mathlib.MeasureTheory... |
5,641 | lean_file_chunk | 3328ec12aac43577fbf10ed575ccb95fa4875dc36a4ab90bbf8b313ddfa45582 | 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, Sébastien Gouëzel, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Integral.Bochner.Set
public import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
public... |
23,985 | lean_file_chunk | 4c72604f7bc416f53f60336becb84968129c04327e64efd34abd79e74d95e0c4 | 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, Sébastien Gouëzel
-/
module
public import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
public... |
19,655 | lean_file_chunk | 1ca057300fe0e947fd04224ec360bbbc3e0474c0175513a5d298c909b2a2b738 | lean_source_file | pkg_mathlib_source_file_chunks | μ.real (closedBall x r) = r ^ finrank ℝ E * μ.real (closedBall 0 1) := by
simp only [measureReal_def, addHaar_closedBall' μ x hr, ENNReal.toReal_mul, mul_eq_mul_right_iff,
ENNReal.toReal_ofReal_eq_iff]
left
positivity
theorem addHaar_unitClosedBall_eq_addHaar_unitBall :
μ (closedBall (0 : E) 1) = μ (ball... |
23,957 | lean_file_chunk | 9dc23449fe7846a870a928a1275f95222b4e8b509b88e90e242aae9a21f0e01c | 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, Sébastien Gouëzel, Yury Kudryashov
-/
module
public import Mathlib.Dynamics.Ergodic.MeasurePreserving
public import Mathlib.LinearAlgebra.Determinant
public import Mat... |
7,546 | lean_file_chunk | 4d641c535b94c331a60d60be1367655701a52f6496fe1668e86a926499fb594f | lean_source_file | pkg_mathlib_source_file_chunks | volume_regionBetween_eq_lintegral' hf.measurable_mk hg.measurable_mk hs]
convert h₂ using 1
· rw [Measure.restrict_prod_eq_prod_univ]
exact (Measure.restrict_eq_self _ (regionBetween_subset f g s)).symm
· rw [Measure.restrict_prod_eq_prod_univ]
exact
(Measure.restrict_eq_self _
(regionBetw... |
23,031 | lean_file_chunk | 71151a61db049354fb7977205f997f0d713280f2ba1cbb3b4a4aa969e68fcafa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
public import Mathlib.LinearAlgebra.Complex.FiniteDimensional
public import Mathlib.Me... |
2,562 | lean_file_chunk | f7b1034bd658f4ad6abf5b527c4bd827fc902523c54c50f1ef0d1f20ab67d7ae | 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.Haar.InnerProductSpace
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
/-!
# Lebesgue meas... |
6,916 | lean_file_chunk | bca76c801a3a572fd5b436bcc20a42fbed2d7df634421364ad1b5751c16f440a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Thomas Zhu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Zhu, Etienne Marion
-/
module
public import Mathlib.Analysis.Calculus.Taylor
public import Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic
import Mathlib.Analysis.Fouri... |
23,934 | lean_file_chunk | d19f22bb3ae3d7b7e8cf9e84728b64976672415b0a39fd875133f4ce4c862029 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jakob Stiefel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Stiefel, Rémy Degenne, Thomas Zhu
-/
module
public import Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar
public import Mathlib.Analysis.Fourier.FourierTransform
public im... |
2,222 | lean_file_chunk | 514a0a9e40af0c48fb084d32a126aac7efda5d15fe63ca0c2ef6efd4c32ae2e1 | lean_source_file | pkg_mathlib_source_file_chunks | [∀ i, NormedAddCommGroup (E i)] [∀ i, NormedSpace ℝ (E i)] {mE : ∀ i, MeasurableSpace (E i)}
[∀ i, BorelSpace (E i)] [∀ i, SecondCountableTopology (E i)] [∀ i, CompleteSpace (E i)]
{μ : (i : ι) → Measure (E i)} {ν : Measure (Π i, E i)} [∀ i, IsFiniteMeasure (μ i)]
[IsFiniteMeasure ν] :
(∀ L, charFunDual... |
23,976 | lean_file_chunk | e8ef0f9146f3ba4780883ad7cb0e8239749e7a2862dec193d08f75bbd7c08a59 | 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.Measure.Restrict
/-!
# Classes for finite measures
We introduce the following typeclasses for measures:
... |
5,301 | lean_file_chunk | c4409a53d8d2ddc40d3354aa595a2c041a50d46f385e3edc270813f1832961c8 | lean_source_file | pkg_mathlib_source_file_chunks | (measure_union_le _ _).trans_lt <| ENNReal.add_lt_top.2 ⟨hU, hV⟩⟩
· intro x hx
rcases (hμ x hx).exists_mem_basis (nhds_basis_opens _) with ⟨U, ⟨hx, hUo⟩, hU⟩
exact ⟨U, nhdsWithin_le_nhds (hUo.mem_nhds hx), U, Subset.rfl, hUo, hU⟩
/-- If `s` is a compact set and `μ` is a locally finite measure, then `s` admit... |
5,340 | lean_file_chunk | f0a6f78a32fc109616981575573363b423abcd8e95f5bac42815243c441aed90 | 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.Measure.Restrict
public import Mathlib.Topology.DiscreteSubset
/-!
# Measures having no atoms
A measure ... |
10,339 | lean_file_chunk | 8ae01bd277b57b635d0adc9250edbf53b1930eb059902e9c2afa5f699eaa4653 | 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.Measure.Typeclasses.Finite
public import Mathlib.Topology.UnitInterval
/-!
# Classes for probability meas... |
23,940 | lean_file_chunk | 12b2e14e8e0a9440d0385f93106caf4c732d2fb815efdcf56bbb89ea903ae129 | 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.Measure.Typeclasses.Finite
/-!
# Classes for s-finite measures
We introduce the following typeclasses fo... |
13,052 | lean_file_chunk | 973657356ca3952b82220642f6f6f31df7f27906d16587ed5ae1cf2b69e2f8a6 | lean_source_file | pkg_mathlib_source_file_chunks | measure_toMeasurable_inter_of_sum hs (fun _ ↦ measure_ne_top _ t) (sum_sfiniteSeq μ).symm
@[simp]
theorem restrict_toMeasurable_of_sFinite [SFinite μ] (s : Set α) :
μ.restrict (toMeasurable μ s) = μ.restrict s :=
ext fun t ht => by
rw [restrict_apply ht, inter_comm t, measure_toMeasurable_inter_of_sFinite ht... |
8,722 | lean_file_chunk | 2b333e3acf60e0fb6a119e872fb52231f2712d296ec403c209d2b0d2419e06fb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Loic Simon
-/
module
public import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
/-!
# Unsigned Hahn decomposition theorem
This file proves the unsigned version o... |
23,982 | lean_file_chunk | 6b86252ae3205d0109ff691daa1145f521fac75fca64ce00b97acab9411636ec | 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, Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
public import Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue
/-!
# Rad... |
10,394 | lean_file_chunk | 770d9de28b2ce219a3148a10f260a3e3f041978cebf95341517ee07a58b80776 | lean_source_file | pkg_mathlib_source_file_chunks | ENNReal.mul_inv_cancel, one_mul]
· simp
· simp [ha_lt_top.ne]
lemma rnDeriv_div_rnDeriv_eq_div_rnDeriv_add {ξ : Measure α}
[SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ξ]
(hμ : μ ≪ ξ) (hν : ν ≪ ξ) :
(fun x ↦ μ.rnDeriv ξ x / ν.rnDeriv ξ x)
=ᵐ[μ + ν] fun x ↦ μ.rnDeriv (μ + ν) x / ν.rnDeriv (μ + ν) ... |
16,267 | lean_file_chunk | d1322a4e020ac6a94fc04e9ee9a9faa062e93b889bd109b0ffc94a0d9248bcbf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Typeclasses.SFinite
/-!
# Method of exhaustion
If `μ, ν` are two measures with `ν` s-finite, then there exists a set... |
10,268 | lean_file_chunk | 43ca91698f7f4aa35d2b3388fc05669900458a1c21460427c2d73f83bd1171c9 | 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, Lorenzo Luccioli
-/
module
public import Mathlib.Analysis.Convex.Continuous
public import Mathlib.Analysis.Convex.Integral
public import Mathlib.MeasureTheory.Measure.Deco... |
23,981 | lean_file_chunk | 868c29a0243a1b28f87b5b845e3dc3ee329e2572f8bfb26af1c3cf495aaa0e66 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
module
public import Mathlib.MeasureTheory.Measure.Decomposition.Hahn
public import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
public import Mathlib.MeasureTheory.Meas... |
23,973 | lean_file_chunk | 825254b60499f02d5f54363da969b86b97678aef5e7666b45ec0310bdca546fb | lean_source_file | pkg_mathlib_source_file_chunks | measurable function such that `μ = s + fν`, then `f = μ.rnDeriv ν`.
This theorem provides the uniqueness of the `rnDeriv` in the Lebesgue decomposition
theorem, while `MeasureTheory.Measure.eq_singularPart` provides the uniqueness of the
`singularPart`. Here, the uniqueness is given in terms of the measures, while the... |
6,490 | lean_file_chunk | eb2ca7d629034cf8d34c9e1e222cf57f5186856937777f5a3f6e3f879ec11d9a | lean_source_file | pkg_mathlib_source_file_chunks | refine (mutuallySingular_singularPart ..).add_right (.singularPart ?_ _)
refine ⟨(s n)ᶜ, (hsm n).compl, ?_⟩
simp [hsm]
-- Finally, the sum of all `ξ n` and measure `ν` with the density `∑' n, f n`
-- is equal to `μ`, thus `(Measure.sum ξ, ∑' n, f n)` is a Lebesgue decomposition for `μ` and `ν`.
have hadd ... |
21,964 | lean_file_chunk | 6f2f49f06da2ef75f4d954b02be1629b2e4e0a22abf5ed00891ba601f6d2b34f | 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.MeasureTheory.Constructions.BorelSpace.Real
public import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
/-!
# Monotone ... |
1,122 | lean_file_chunk | c6a0977c9d66920685372df380bb7d8ec3a3e2eb980712c4a0646771b04d0ea4 | 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.Analysis.Normed.Group.Real
public import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
/-!
# Interactions between the L... |
11,174 | lean_file_chunk | de5a969789bcd3761bcc8c730d0a9b2ed945704078e5727894d16741028798f1 | 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.MeasureTheory.Integral.Lebesgue.Markov
public import Mathlib.MeasureTheory.Integral.Lebesgue.Sub
/-!
# Dominated conver... |
6,233 | lean_file_chunk | 0a18f168f47374634851b714f10db2f8a27ecd994bedc69b3284c47337acacd6 | 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.Dynamics.Ergodic.MeasurePreserving
public import Mathlib.MeasureTheory.Integral.Lebesgue.Add
/-!
# Behavior of the Lebe... |
23,951 | lean_file_chunk | 22a44a0dc03af61e889304b5d36232976d945947765994c7f1c33af95fb2aa95 | 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.MeasureTheory.Function.SimpleFunc
public import Mathlib.Algebra.Order.Pi
/-!
# Lower Lebesgue integral for `ℝ≥0∞`-value... |
8,152 | lean_file_chunk | 00708c1ca8a8f2aebeec1591271fffabc69c51b473c85d3e071de61ae82d64ed | lean_source_file | pkg_mathlib_source_file_chunks | (lintegral_indicator_le _ _).trans (setLIntegral_const s c).le
theorem lintegral_indicator_const₀ {s : Set α} (hs : NullMeasurableSet s μ) (c : ℝ≥0∞) :
∫⁻ a, s.indicator (fun _ => c) a ∂μ = c * μ s := by
rw [lintegral_indicator₀ hs, setLIntegral_const]
theorem lintegral_indicator_const {s : Set α} (hs : Measura... |
7,424 | lean_file_chunk | fbdae758f6fba15c3842806b75852dee9da77e9f63e8278f08087f272082d7f8 | 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.MeasureTheory.Integral.Lebesgue.Add
/-!
# Markov's inequality
The classical form of Markov's inequality states that fo... |
9,243 | lean_file_chunk | e7d09f94b3aa07ad8e28462ebb980bd483e8e595361cafce8efbe865e155cbd2 | 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.MeasureTheory.Integral.Lebesgue.Add
/-!
# Subtraction of Lebesgue integrals
In this file we first show that Lebesgue i... |
17,630 | lean_file_chunk | efda4593d01184f5247635fd6d4c66cac9dfc2ddc6d1aa8538cae82e504f4da8 | 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.MeasureTheory.Integral.Lebesgue.Map
public import Mathlib.MeasureTheory.Integral.Lebesgue.Markov
public import Mathlib.M... |
15,091 | lean_file_chunk | e791c5ff8e5a1e07d4154627682bde644d80030102dfd1087b9a9cb72c27c9f7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jesse Reimann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jesse Reimann, Kalle Kytölä
-/
module
public import Mathlib.MeasureTheory.Measure.Content
public import Mathlib.Topology.ContinuousMap.CompactlySupported
public import Mathlib.Topology... |
4,213 | lean_file_chunk | 2c3f1bed8025067d9717c85f19d4d3b0406b1efb99f34da0c303ef46e0770299 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yoh Tanimioto. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yoh Tanimoto
-/
module
public import Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real
/-!
# Riesz–Markov–Kakutani representation theorem for `ℝ≥0`
This file proves the Riesz-M... |
23,948 | lean_file_chunk | c8856dde4f5fd093f431f3c0248de72917c85cfe12308ee9f46e815afc674fff | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yoh Tanimoto. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yoh Tanimoto, Oliver Butterley
-/
module
public import Mathlib.MeasureTheory.Integral.Bochner.Set
public import Mathlib.MeasureTheory.Integral.CompactlySupported
public import Mathlib.M... |
196 | lean_file_chunk | ae9196b42161cc1c605796e2f897f75ad0200049e8588c5c74fa0662e2e18f38 | lean_source_file | pkg_mathlib_source_file_chunks | simp only [map_comap_subtype_coe hK.measurableSet, μ', Measure.restrict_eq_self_of_ae_mem h]
· rw [integral_map (φ := Subtype.val) (by fun_prop) (by fun_prop)]
simp
end Compact
end RealRMK |
20,011 | lean_file_chunk | 6195adfeacf243739a2bc91a8c58e1e35f810daa7096ff89133a2bbf188cb9aa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.Calculus.Deriv.Prod
public import Mathlib.Analysis.Calculus.DiffContOnCl
public import Mathlib.Analysis.Calculus.FDeriv.Symme... |
23,995 | lean_file_chunk | 53528ea783c6ec57fc30e1e917be7d0d05d3963b338a7c8d4d1c9804ad3163e8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 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.Calculus.ContDiff.Deriv
public import Mathlib.Analysis.Calculus.Deriv.Aff... |
490 | lean_file_chunk | fcf49c6d4be78f7d07b51869c52428dfa7ec4b685aab2bb69e9230e9dbfd5c43 | lean_source_file | pkg_mathlib_source_file_chunks | (by simpa only [nhdsWithin_univ, continuousWithinAt_univ]) (mem_univ _) |>.hasFDerivAt_of_univ
/-- The integral of `ω` along `[a -[ℝ] b]`, as a function of `b`, has derivative `ω a` at `b = a`.
This is a `HasFDerivAt` version assuming that `ω` is continuous on the whole space. -/
theorem HasFDerivAt.curveIntegral_segm... |
10,219 | lean_file_chunk | b399f7eba22ff8f222aa4c6aec650ef5447a7ddd6d30b5ed797f28f65bc85982 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
module
public import Mathlib.Analysis.Calculus.Deriv.Basic
public import Mathlib.Analysis.Calculus.DiffContOnCl
public import Mathlib.MeasureTheory.Integral.... |
4,058 | lean_file_chunk | e1602a04f34df90a42f4ccac0ef71927d9ce1e02b96e6e8795911d73c0bd15eb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yizheng Zhu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yizheng Zhu
-/
module
public import Mathlib.Analysis.Calculus.Deriv.Slope
public import Mathlib.MeasureTheory.Covering.OneDim
public import Mathlib.MeasureTheory.Integral.IntervalIntegra... |
23,913 | lean_file_chunk | 19d18389e4ef569a313ebb00326c392d9dc986166916db5c455523bc068995c1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Function.JacobianOneDim
public import Mathlib.MeasureTheory.Integral.IntervalIntegral... |
6,054 | lean_file_chunk | 119968f0971a0818b5265579b3ffb122b8d06203083660a49219616e122db49a | lean_source_file | pkg_mathlib_source_file_chunks | (hg' : Continuous g') : (∫ x in a..b, f' x • (g' ∘ f) x) = (g ∘ f) b - (g ∘ f) a :=
integral_eq_sub_of_hasDerivAt (fun x hx ↦ (hg x hx).scomp x <| hf x hx)
(hf'.smul (hg'.comp_continuousOn <| HasDerivAt.continuousOn hf)).intervalIntegrable
@[deprecated (since := "2026-03-19")]
alias integral_deriv_comp_smul_deri... |
5,180 | lean_file_chunk | 6a52db88929c261dfaaa5ee5a086aab7ce0d46a1a0a09c3ddf19486695f235ac | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Calculus.ContDiff.Deriv
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
/-! # Fundamental ... |
23,997 | lean_file_chunk | e7265a52cb502fd0dd3dc2a3691978ad28bd18835c184ef0c544f7b445208f59 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
public import Mathlib.MeasureTheory.Measure.Lebe... |
23,988 | lean_file_chunk | cead6b47fed600e637be6dccb47131fe0bbe8e83c2d33e46d38965592ef9d0f3 | lean_source_file | pkg_mathlib_source_file_chunks | /-- An even function is interval integrable (with respect to the volume measure) on every interval
if it is interval integrable (with respect to the volume measure) on every interval of the form
`0..x`, for positive `x`. -/
theorem intervalIntegrable_of_even
(h₁f : ∀ x, f x = f (-x)) (h₂f : ∀ x, 0 < x → IntervalInt... |
16,832 | lean_file_chunk | 0c27b04475d2f1a1b60a55aed8e682df061250afa28e2be3d97971c119cf3164 | lean_source_file | pkg_mathlib_source_file_chunks | sub_eq_of_eq_add' <| Eq.symm <| integral_add_adjacent_intervals hac (hac.symm.trans hab)
theorem integral_interval_add_interval_comm (hab : IntervalIntegrable f μ a b)
(hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c) :
((∫ x in a..b, f x ∂μ) + ∫ x in c..d, f x ∂μ) =
(∫ x in a..d, f x... |
3,753 | lean_file_chunk | a6a2ff52d63154249fb32c64bd80309d2cba6041f34947440fde0ef4a5ef5790 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yizheng Zhu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yizheng Zhu
-/
module
public import Mathlib.LinearAlgebra.AffineSpace.Slope
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
/-!
# Some properties of the interval int... |
23,985 | lean_file_chunk | bbe3ae30e095dc9df0f63888d503b85ff1b1bb152366d7d025a06e014586e2be | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Calculus.Deriv.Add
public import Mathlib.Analysis.Calculus.Deriv.Comp
public import Mathli... |
23,961 | lean_file_chunk | ce85f8f6cc580c5824717af98c3a0dd28e8c57b0355a06c75ec2da6be8fa3cc1 | lean_source_file | pkg_mathlib_source_file_chunks | {f : ℝ → E} {c ca cb : E} {l l' la la' lb lb' : Filter ℝ} {lt : Filter ι} {a b : ℝ}
{u v ua ub va vb : ι → ℝ} [FTCFilter a la la'] [FTCFilter b lb lb']
/-!
#### Auxiliary `Asymptotics.IsLittleO` statements
In this section we prove several lemmas that can be interpreted as strict differentiability of
`(u, v) ↦ ∫ x i... |
23,991 | lean_file_chunk | 118a19b2d16f8a647273df198d995dc925935b7cc529cf3b4bfbcc250e865242 | lean_source_file | pkg_mathlib_source_file_chunks | (resp., left) derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `c`. -/
theorem derivWithin_integral_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b)
{s t : Set ℝ} [FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b))
(hb : Tendsto f (𝓝[t] b ⊓ ae volume) (𝓝 c))
(hs ... |
656 | lean_file_chunk | ca7c940b28ea72f4a64e398fee4ac32cd3d2e09d87f3cb079ae07a7cd732759d | lean_source_file | pkg_mathlib_source_file_chunks | (hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x)
(hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x) : IntervalIntegrable g' volume a b := by
rcases le_total a b with hab | hab
· simp only [uIcc_of_le, min_eq_left, max_eq_right, IntervalIntegrable, hab,
Ioc_eq_empty_of_le, integrableOn_empty, ... |
14,155 | lean_file_chunk | 51870a68b61013ace3578f9c978f814ac93af2a0db8947cdbe7d2663319f04ff | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yizheng Zhu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yizheng Zhu
-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Gaps
public import Mathlib.Analysis.Calculus.Deriv.Mul
public import Mathlib.MeasureTheory.Integral.Interval... |
8,722 | lean_file_chunk | e3fec8f05ed2a6cd758ea7f1eae8c8958c7f7494b442fd63c204c0736771f6a6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yizheng Zhu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yizheng Zhu
-/
module
public import Mathlib.Analysis.BoundedVariation
public import Mathlib.MeasureTheory.Function.AbsolutelyContinuous
public import Mathlib.MeasureTheory.Integral.Inter... |
22,791 | lean_file_chunk | 24cd5d725778ba5b8adaf1f72f9d17e38b59d21744b61997e32bac54badc9e40 | 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.Measure.Lebesgue.EqHaar
public import Mathlib.MeasureTheory.Measure.Haar.Quotient
pub... |
3,621 | lean_file_chunk | a44cb0f025c35badcb01abb347cbe113db8279461138a33edaa856cfcf0e427e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Louis (Yiyang) Liu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis (Yiyang) Liu
-/
module
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
public import Mathlib.MeasureTheory.Integral.MeanValue
/-!
# First mean value the... |
14,125 | lean_file_chunk | 8a192981f10bd2b674211394820e2646bcffb2cb73a889f709ea7c92809b0568 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 P. Michael Kielstra. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: P. Michael Kielstra
-/
module
public import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
public import Mathlib.Tactic.Field
/-!
# The trapezoidal rule
This file contains a... |
16,879 | lean_file_chunk | bff468053f06924eb12a54e01150864b2ab4124d138938d755ac0156600954bb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
module
public import Mathlib.Analysis.Normed.Operator.CompleteCodomain
public import Mathlib.MeasureTheory.Integral.Bochner.Set
public import Mathlib.... |
9,568 | lean_file_chunk | 5e9697d74ec68f2d71adcfce54a376a34bf93720af2e80c82b51165600736d22 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 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.Basic
import Mathlib.Analysis.Normed.Module.FiniteDimension
/-!
# Integral with respect to a sum of mea... |
23,980 | lean_file_chunk | 033a4d0a21fbd2e13c2e90c36c130c2c69b79a63cce92f3ff97526948c3ac86a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Group.MeasurableEquiv
public import Mathlib.MeasureTheory.Integral.Bochner... |
23,928 | lean_file_chunk | 5f4b10d674bffa7a66c565e94d41cd904eda5b9293fa960b2ccd2d3e9423bccb | lean_source_file | pkg_mathlib_source_file_chunks | · filter_upwards; simp_rw [Pi.zero_apply, norm_nonneg, imp_true_iff]
theorem ofReal_integral_norm_eq_lintegral_enorm {P : Type*} [NormedAddCommGroup P] {f : α → P}
(hf : Integrable f μ) : ENNReal.ofReal (∫ x, ‖f x‖ ∂μ) = ∫⁻ x, ‖f x‖ₑ ∂μ := by
rw [integral_norm_eq_lintegral_enorm hf.aestronglyMeasurable, ENNReal.... |
19,491 | lean_file_chunk | 589e4a064b8e507839ce7f978b475ba2e37ab5893ec30c7a3869f3cfc6672da8 | lean_source_file | pkg_mathlib_source_file_chunks | lemma integral_of_isEmpty [IsEmpty α] {f : α → G} : ∫ x, f x ∂μ = 0 :=
μ.eq_zero_of_isEmpty ▸ integral_zero_measure _
theorem integral_finset_sum_measure {ι} {m : MeasurableSpace α} {f : α → G} {μ : ι → Measure α}
{s : Finset ι} (hf : ∀ i ∈ s, Integrable f (μ i)) :
∫ a, f a ∂(∑ i ∈ s, μ i) = ∑ i ∈ s, ∫ a, f ... |
6,946 | lean_file_chunk | 0b87b3b30573b0f1bf697a2b0dc9bba3e7d93541f4a7d2fabce7ea843ec89ac4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Integral.Bochner.Set
/-!
# Fundamental theorem of calculus for set integrals
This file proves a version o... |
23,990 | lean_file_chunk | 2c6c29bca1416619b66b414d07367a762eabab7b36f86495bc04cf05f4e76128 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
module
public import Mathlib.Combinatorics.Enumerative.InclusionExclusion
public import Mathlib.MeasureTheory.Function.LocallyIntegrable
public import... |
23,965 | lean_file_chunk | ce2006787d73b3a4f6cb75271ec3581ea3f7e48ec0662118f4ea3045f401f230 | lean_source_file | pkg_mathlib_source_file_chunks | rw [← integral_add_compl₀ h_meas hfi.norm]
_ = ∫ x in {x | 0 ≤ f x}, f x ∂μ + ∫ x in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ := by
congr 1
refine setIntegral_congr_fun₀ h_meas fun x hx => ?_
dsimp only
rw [Real.norm_eq_abs, abs_eq_self.mpr _]
exact hx
_ = ∫ x in {x | 0 ≤ f x}, f x ∂μ - ∫ x in {x ... |
8,449 | lean_file_chunk | 4c293ecd36d8b5fd29687deecaa884b48290d763c5b83654167c2f6ac844f0b2 | lean_source_file | pkg_mathlib_source_file_chunks | theorem MeasureTheory.setIntegral_support : ∫ x in support F, F x ∂ν = ∫ x, F x ∂ν := by
nth_rw 2 [← setIntegral_univ]
rw [setIntegral_eq_of_subset_of_forall_diff_eq_zero MeasurableSet.univ (subset_univ (support F))]
exact fun _ hx => notMem_support.mp <| notMem_of_mem_diff hx
theorem MeasureTheory.setIntegral_t... |
23,998 | lean_file_chunk | eb7e7f8617d475573496f92421e1a6cff40a2dc0211010876574dedc1898ec0c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Measure.Regular
public import Mathlib.Topology.Semicontinuity.Basic
public import Mathlib.MeasureTheory.Integral.Boc... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.