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 |
|---|---|---|---|---|---|
21,941 | lean_file_chunk | 25ffe1fb38392b85a8015c5b1d20b9dbc2131cef3b346dbd9ca8e55c1b238972 | 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.Order.SuccPred.IntervalSucc
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basi... |
14,233 | lean_file_chunk | 3f9a84a1a01aeacc6aaced03432ef8d6c8187a9e9fd94e0b415c0a81026ac5a6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 David Ledvinka. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Ledvinka
-/
module
public import Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
import Mathlib.Probability... |
13,527 | lean_file_chunk | 51f16d46cad9dba5a5a6869f1d3c9a71498f504a87696d150bf453197a03fc92 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.Analysis.InnerProductSpace.LinearMap
public import Mathlib.Analysis.RCLike.Lemmas
public import Mathlib.MeasureTheory.Function.LpSpace.Conti... |
5,576 | lean_file_chunk | d8e9a329593443a2c2f1f0494c1c32c098271354758d49d22195931633c9a246 | 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.Constructions.BorelSpace.Order
/-!
# Measurability criterion for ennreal-valued functions
Consider a function `f :... |
23,947 | lean_file_chunk | 84957cb08512182c3c4cfb523c1cc0972d464acd0c54267d25e948063530ec06 | 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.Deriv.MeanValue
public import Mathlib.MeasureTheory.Function.Jacobian
/-!
# Change of variable formulas for int... |
5,337 | lean_file_chunk | 804b95543fe5b2697b3c235bf502584154fdb3d63ce93bd434415cb8bc1f8010 | lean_source_file | pkg_mathlib_source_file_chunks | have B : ∫ x in n '' (e '' s), g x = ∫ x in e '' s, (|-1| : ℝ) • g (n x) :=
integral_image_eq_integral_abs_deriv_smul (hs.image_of_monotoneOn hf.neg)
(fun x hx ↦ hasDerivWithinAt_neg _ _) neg_injective.injOn _
simp only [abs_neg, abs_one, one_smul] at B
rw [A, ← image_comp] at B
convert B using 3 with x... |
23,992 | lean_file_chunk | 1f45888c756fce929adfe8c678db0175df834a444b7e781816a7024542fbdce8 | 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.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.L... |
13,445 | lean_file_chunk | 7ca406e34654e33a236e02e1b9c19295f4f97266b44097529c209b29cbb5fa01 | lean_source_file | pkg_mathlib_source_file_chunks | [LinearOrder X] [CompactIccSpace X] [NoMinOrder X] [OrderTopology X] :
IntegrableOn f (Iio a) μ ↔ IntegrableAtFilter f atBot μ ∧
IntegrableAtFilter f (𝓝[<] a) μ ∧ LocallyIntegrableOn f (Iio a) μ := by
constructor
· intro h
exact ⟨⟨Iio a, Iio_mem_atBot a, h⟩, ⟨Iio a, self_mem_nhdsWithin, h⟩, h.locallyIn... |
3,001 | lean_file_chunk | 86a8b8d23b63b417aac80f982fee988757d80ba3f31ddc0c4426cd0fa8d84444 | 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.Constructions.BorelSpace.Order
/-!
# Measurability of `⌊x⌋` etc
In this file we prove that `Int.floor`, `Int.ceil`, `I... |
23,990 | lean_file_chunk | 10d534dec8b1e0cfce3fa927d58c7bf45d7d8fec60c63dfddfd8442412511a87 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
module
public import Mathlib.MeasureTheory.Function.L1Space.AEEqFun
public import Mathlib.MeasureTheory.Function.LpSpace.Complete
pub... |
16,656 | lean_file_chunk | 49323977004a89bd431f4010d48f908c93adff072ac828d8f06c804bd36edb70 | lean_source_file | pkg_mathlib_source_file_chunks | repeat intro h; rw [h]
theorem sub_toSimpleFunc (f g : Lp.simpleFunc E p μ) :
toSimpleFunc (f - g) =ᵐ[μ] toSimpleFunc f - toSimpleFunc g := by
filter_upwards [toSimpleFunc_eq_toFun (f - g), toSimpleFunc_eq_toFun f,
toSimpleFunc_eq_toFun g, Lp.coeFn_sub (f : Lp E p μ) g] with _
simp only [AddSubgroup.coe_su... |
11,267 | lean_file_chunk | 9afc03b0b26ee39acc07e66f21471a53f2dc427328714b1d673e426bf6220fb2 | 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, Heather Macbeth
-/
module
public import Mathlib.MeasureTheory.Function.SimpleFunc
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizabl... |
11,394 | lean_file_chunk | 606285f0c3119f65de2daf907212b213b7d07fc42134b9bb6f71602e984c7482 | 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.StronglyMeasurable.Basic
/-!
# Egorov theorem
This file contains the Egorov theorem which states that an almost every... |
21,846 | lean_file_chunk | f16d7e503c2410f22eba3d7d37e0de5eb07f1d94cd633ad9458d36b28e703e8c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.Analysis.InnerProductSpace.Continuous
public import Mathlib.Analysis.Normed.Module.HahnBanach
public import Mathlib.MeasureTheory.Function.A... |
5,962 | lean_file_chunk | ef6469c9a7b4b030abf078ba8a24ddf31d39d843336185336df8e32e93e4bd23 | 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.MeasurableSpace.Basic
public import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Sequence of measurable functions asso... |
16,667 | lean_file_chunk | 62591b90d6fd462a8957fda4e18eafe8f8c45bbf4e263946ba0ee745872e2b7f | 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.Measure.Portmanteau
public import Mathlib.Probability.IdentDistrib
/-!
# Convergence in distribution
We introduce a definiti... |
6,767 | lean_file_chunk | 7962f1e8f6b1f349a179c55c39a6d086704fa03596231e579349772c44e014a1 | 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.Typeclasses.SFinite
/-!
# Restriction of a measure to a sub-σ-algebra
## Main definitions
* `MeasureTheory.Measure... |
10,905 | lean_file_chunk | f9b80f202bbaaf667a49cd57e8f3a61b049f1eebdadb45741ae0feb40a2708c6 | 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, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Measure.Restrict
/-! # Mutually singular measures
Two measures `μ`, `ν` are said to be mutually singular (`Me... |
6,642 | lean_file_chunk | c9afb7b8edae1d6039949009fc38741389efe3bf754d249fae02289e10ca3e03 | 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.Dirac
public import Mathlib.Topology.Algebra.InfiniteSum.ENNReal
/-!
# Counting measure
In this file we define t... |
23,970 | lean_file_chunk | 354c5f31246bbf6a433c3df041adecf7eb15dc4ab1c7e5508a4bc7dd51a9d11c | 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.Function.SimpleFuncDenseLp
public import Mathlib.MeasureTheory.SetAlgebra
/-!
# Separable measure
The goal of this file ... |
4,878 | lean_file_chunk | 3d552025e99c66fba3b52d8c179ae819cf34d450bf798ac078713c446b24a871 | lean_source_file | pkg_mathlib_source_file_chunks | -- in `u` and `𝒜₀` associates the corresponding sum.
let key (n : ℕ) (d : Fin n → u) (s : Fin n → 𝒜₀) : (Lp E p μ) :=
∑ i, indicatorConstLp p (h𝒜₀.measurable (s i) (Subtype.mem (s i))) (s i).2.2 (d i : E)
let D := {s : Lp E p μ | ∃ n d t, s = key n d t}
refine ⟨D, ?_, ?_⟩
· -- Countability directly follo... |
5,183 | lean_file_chunk | 152bf9a0a3088603e75e4838d900708b92f003ec0df97874c5494a5f71193405 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
/-!
# Products of finite measures and probability measures
This file introduces finite... |
16,079 | lean_file_chunk | 0e80cc39edffab324f5f6c456d142756a5ebfa642d2ed545b3ad8209c1dceb01 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
/-!
# Exponentially tilted measures
The exponential tilting of a measure `μ` on `α` by a ... |
8,698 | lean_file_chunk | 6a6b42613373400dafc86bd58a1e9704d18693afeb9378ec4185b3a2f44da4b8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.Topology.Separation.CompletelyRegular
public import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
/-!
# Dirac deltas as probability meas... |
5,509 | lean_file_chunk | 80bf8535e4e859e097b6f4e12064cd5dd582df3d605f2653d9ebfe93183af880 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Almost everywhere disjoint sets
We say that sets `s` and `t` are `μ`-a.e. disjoint (see ... |
23,933 | lean_file_chunk | b34e6791ccd804f65af1d830b8b3d4042d27fab0202627cdf15b7b6c6cd00758 | 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, Floris van Doorn, Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
public import Mathlib.MeasureTheory.Group.Measurab... |
23,938 | lean_file_chunk | 71fad6fd36089326f06ed64a8f6ff7f23599c6bc21c95790eba5f339e392d061 | lean_source_file | pkg_mathlib_source_file_chunks | _ = μ (⋃ n, A n) + ∑' n, δ n := (congr_arg₂ (· + ·) (measure_iUnion hAd hAm).symm rfl)
_ < r := hδε
/-- See also `IsCompact.measure_closure` for a version
that assumes the `σ`-algebra to be the Borel `σ`-algebra but makes no assumptions on `μ`. -/
lemma measure_closure_eq_of_isCompact [R1Space α] [OuterRegular μ]
... |
13,046 | lean_file_chunk | 2078fb2388cd85122b9a2f4bc091ede400a4b01e9a99a6eca9c2669611c833fc | lean_source_file | pkg_mathlib_source_file_chunks | See also `Set.exists_isOpen_lt_of_lt` and `MeasurableSet.exists_isOpen_diff_lt`
for the case of an outer regular measure. -/
protected theorem _root_.MeasurableSet.exists_isOpen_symmDiff_lt [InnerRegularCompactLTTop μ]
[IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α]
{s : Set α} (hs : MeasurableSet s) (hμs... |
6,279 | lean_file_chunk | 7c283dc19933df3c0a7d47bdca583a36b5f2d85e6bc2bae55a320a79c3f47be6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
module
public import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Uniformly locally doubling measures
A uniformly locally doubling measure `μ` on a metric space i... |
8,297 | lean_file_chunk | d042e9603e78bce9cfe4b801040dd23d096bef41e0fd40dcd1e27deea04f1436 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jon Bannon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Bannon, Jireh Loreaux
-/
module
public import Mathlib.MeasureTheory.Measure.OpenPos
/-!
# Support of a Measure
This file develops the theory of the **support** of a measure `μ` on a... |
8,779 | lean_file_chunk | c404b3af432e27b56e575c74dc0346801a4a1711ca496438bfe04f59a339ee10 | 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, Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Pullback of a measure
In this file we define the pullback `MeasureT... |
23,941 | lean_file_chunk | e0ccef6ca738b1afce48bf4c00362134edc7dd9301fbd893682493bfd764efaa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.MeasureTheory.Measure.FiniteMeasure
public import Mathlib.MeasureTheory.Integral.Average
public import Mathlib.MeasureTheory.Measure.Prod
/... |
7,856 | lean_file_chunk | f9a924e4814fc0c11979bae23590d397d15344d0fa96436f05e746051bcbfd0f | lean_source_file | pkg_mathlib_source_file_chunks | (mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)) (f : Ω →ᵇ ℝ≥0) :
Tendsto (fun i ↦ (μs i).testAgainstNN f) F (𝓝 (μ.testAgainstNN f)) := by
by_cases h_mass : μ.mass = 0
· simp only [μ.mass_zero_iff.mp h_mass, zero_testAgainstNN_apply, zero_mass] at mass_lim ⊢
exact tendsto_zero_testAgainstNN_of_tend... |
4,712 | lean_file_chunk | 0217141666d30740c7b9559833aab6c3bd67647569c40f97c65f8ab1f7f7bb2c | 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.Decomposition.Exhaustion
public import Mathlib.Probability.ConditionalProbability
/-!
# s-finite measures can be writ... |
8,276 | lean_file_chunk | c398976f977447245add09955949d48f1138aa871a48169a49521a18a11f009e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
public import Mathlib.MeasureTheory.Function.SpecialFunctions.Sinc
public import Mathlib.MeasureTh... |
12,074 | lean_file_chunk | addf13e205ecf283917069c750f70604411864a9ae89f22a14d0b4c58b1491e0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
public import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
public import Mathlib.Mea... |
13,121 | lean_file_chunk | a85f9a5f5ea3c5da4db6730be0ba6459f34ce0c2d39735784860ac3d3078c27e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.Analysis.InnerProductSpace.PiL2
public import Mathlib.MeasureTheory.Measure.Tight
import Mathlib.MeasureTheory.Constructions.BorelSpace.Ord... |
10,525 | lean_file_chunk | 24b7934321e016979ef868e0034cead77fa87f0595b6dd63902631c639e47111 | 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.Analysis.SpecificLimits.Basic
public import Mathlib.MeasureTheory.Measure.Regular
public import Mathlib.Topology.GDelta.... |
23,947 | lean_file_chunk | e72322efb52c4302a07a2b29e175a4f0dccaaf04c38e150f810db37984a18b3f | 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.MeasurablyGenerated
public import Mathlib.MeasureTheory.Measure.NullMeasurable
public impo... |
23,935 | lean_file_chunk | 0263a9cc5d7f5a529c0eb766104f8dd455dbee8025153d96f16a9ce5c071638d | lean_source_file | pkg_mathlib_source_file_chunks | [(atBot : Filter ι).IsCountablyGenerated] {s : ι → Set α} (hs : Antitone s) :
μ (⋃ i, s i) = ⨆ i, μ (s i) :=
hs.dual_left.measure_iUnion
/-- Continuity from below: the measure of the union of a sequence of
(not necessarily measurable) sets is the supremum of the measures of the partial unions. -/
theorem measure... |
18,680 | lean_file_chunk | b2374a1f9c621fc8b422c647de4ef4a239a328f0b27cca7c4220cf1ea7a81500 | lean_source_file | pkg_mathlib_source_file_chunks | measure_eq_left_of_subset_of_measure_add_eq ?_ (subset_toMeasurable _ _)
(measure_toMeasurable t).symm
rwa [measure_toMeasurable t]
· simp only [not_or, ENNReal.add_eq_top, Pi.add_apply, Ne, coe_add] at ht
exact ht.1
theorem measure_toMeasurable_add_inter_right {s t : Set α} (hs : MeasurableSet s)
... |
23,995 | lean_file_chunk | ecab2a5ed80bfc5f1774bf4f2cbe845f68935d2fa613c68c0333916c2c2dc6aa | 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.GiryMonad
public import Mathlib.MeasureTheory.Measure.OpenPos
public import Mathlib.MeasureTheory.Measure.Doub... |
23,981 | lean_file_chunk | 1067cde6dbcf0936a84162a8b654ab69eb1d0212955445a2518b4cdad63731ff | lean_source_file | pkg_mathlib_source_file_chunks | refine
⟨fun n => hμ.set n.unpair.1 ×ˢ hν.set n.unpair.2, fun n =>
mem_image2_of_mem (hμ.set_mem _) (hν.set_mem _), fun n => ?_, ?_⟩
· rw [prod_prod]
exact mul_lt_top (hμ.finite _) (hν.finite _)
· simp_rw [iUnion_unpair_prod, hμ.spanning, hν.spanning, univ_prod_univ]
lemma prod_sum_left {ι : Type*} (m... |
9,947 | lean_file_chunk | 11ef9227f18a2c353d2e79c1bc6012703d22bb9b3a26b694f0a6b87d9287d6ee | lean_source_file | pkg_mathlib_source_file_chunks | theorem lintegral_prod_le (f : α × β → ℝ≥0∞) :
∫⁻ z, f z ∂μ.prod ν ≤ ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ := by
rw [Measure.prod]
exact (lintegral_bind_le _ _ _).trans <| lintegral_mono fun a ↦ lintegral_map_le _ _
/-- **Tonelli's Theorem for set integrals**: For `ℝ≥0∞`-valued almost everywhere measurable
functions on `... |
13,321 | lean_file_chunk | 92998ec1b36483f13a709b80c52749d541b2b3b93fee5981ca43bc423a31389d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Josha Dekker, Arav Bhattacharyya
-/
module
public import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
public import Mathlib.Topology.Metrizable.CompletelyMetrizable
im... |
23,988 | lean_file_chunk | 75fec527f16d553340eb937dad764fc4e31e40ace8e7f7dd9efd2ce16307484c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
public import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
public import Mathlib.MeasureTheory.Inte... |
19,731 | lean_file_chunk | 7149285cc686d827b810610e2f3f72357b830d92bd9d0d8c4b69723a75d79c30 | lean_source_file | pkg_mathlib_source_file_chunks | [PseudoMetrizableSpace Ω] [OpensMeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
(h : ∀ {E}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i ↦ μs i E) L (𝓝 (μ E)))
(G : Set Ω) (G_open : IsOpen G) :
μ G ≤ L.liminf (fun i ↦ μs i G) ... |
23,969 | lean_file_chunk | 907f1f91905979285f40667b640dc0a94edebecad5a5b3aeab67ac1de2f9d6da | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.Analysis.RCLike.Lemmas
public import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
public import Mathlib.MeasureTheory.Measure.... |
23,940 | lean_file_chunk | 50130754e0e8040157c0c076141efe70790596dcbd2ff328ec35505a1e92087f | lean_source_file | pkg_mathlib_source_file_chunks | IsInducing.tendsto_nhds_iff ⟨rfl⟩
theorem tendsto_iff_forall_toWeakDualBCNN_tendsto {γ : Type*} {F : Filter γ}
{μs : γ → FiniteMeasure Ω} {μ : FiniteMeasure Ω} :
Tendsto μs F (𝓝 μ) ↔
∀ f : Ω →ᵇ ℝ≥0, Tendsto (fun i ↦ (μs i).toWeakDualBCNN f) F (𝓝 (μ.toWeakDualBCNN f)) := by
rw [tendsto_iff_weakDual_te... |
3,869 | lean_file_chunk | 1eaa52e992ac33da8ff8f12168d5f64ae8188bd6cbab04c99af16ef05d29f048 | 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
-/
module
public import Mathlib.Analysis.RCLike.BoundedContinuous
public import Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral
public import Mathlib.MeasureTheo... |
10,961 | lean_file_chunk | d25a38a27a964555ded426c863872b522a7ac33c1a1075eebfb50cd2da45ca3b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Measure.Tilted
/-!
# Log-likelihood Ratio
The likelihood ratio between two measures `μ` and `ν` is their Radon-Nikodym deriv... |
13,104 | lean_file_chunk | 45e955d80208e8bc6f405c7600bbfdc64d7add15f86acde07d9f03078ce1c190 | 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.Embedding
public import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Pushforward of a... |
8,382 | lean_file_chunk | 629624e9a364b039e47edab61c28edb95ba7ffdd0d6cd933eae96be54f841547 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
public import Mathlib.MeasureTheory.Measure.Prod
/-!
# Products of finite measures and probabilit... |
16,599 | lean_file_chunk | c1efce783488f8c33a9825caf169699cec72b906bd7dfbf838b38133abe541ef | 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.MeasurableSpace.CountablyGenerated
public import Mathlib.MeasureTheory.Measure.MutuallySingular
public import Mathlib.Meas... |
23,962 | lean_file_chunk | 7793801a1e938a614b8e87431fba987f741d97c84a794251d4bdf87849dd87e0 | 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.Comap
public import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
public import Mathlib.Dat... |
23,945 | lean_file_chunk | 15d6d0ae49186428bb6de1f148ebc10ac3bd39d60cb58150c9d4a82246c4ad68 | lean_source_file | pkg_mathlib_source_file_chunks | ext fun t ht => by simp only [sum_apply _ ht, restrict_iUnion_apply_ae hd hm ht]
theorem restrict_iUnion [Countable ι] {s : ι → Set α} (hd : Pairwise (Disjoint on s))
(hm : ∀ i, MeasurableSet (s i)) : μ.restrict (⋃ i, s i) = sum fun i => μ.restrict (s i) :=
restrict_iUnion_ae hd.aedisjoint fun i => (hm i).nullMe... |
5,530 | lean_file_chunk | e028881e3b0d74b9512cf8b9b9bef886720c7583820a4e7820bac2d9348cb418 | lean_source_file | pkg_mathlib_source_file_chunks | (hf : f =ᵐ[μ.restrict sᶜ] 0) : s.indicator f =ᵐ[μ] f := by
rw [Filter.EventuallyEq, ae_restrict_iff' hs.compl] at hf
filter_upwards [hf] with x hx
by_cases hxs : x ∈ s
· simp only [hxs, Set.indicator_of_mem]
· simp only [hx hxs, Pi.zero_apply, Set.indicator_apply_eq_zero, imp_true_iff]
theorem indicator_ae_e... |
513 | lean_file_chunk | 9a13a432f5d060855aec9b2a4fc210a5b971da84804b33ff133dc5f669e41fd8 | lean_source_file | pkg_mathlib_source_file_chunks | module
public import Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar
public import Mathlib.Analysis.Fourier.FourierTransform
public import Mathlib.Analysis.InnerProductSpace.Dual
public import Mathlib.Analysis.InnerProductSpace.ProdL2
public import Mathlib.Analysis.Normed.Lp.MeasurableSpace
public import Mathli... |
23,994 | lean_file_chunk | c9df4ea281d112d4104bbc42a068d39f294c29fe62bc3952f29345ca5027d087 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
public import Mathlib.MeasureTheory.Measure.Tight
import Mathlib.MeasureTheory.Integral.R... |
3,811 | lean_file_chunk | f34eb23363143c8e5f73e0303d326c32163bf4702706aa4e9ff2f3adcd1acab2 | lean_source_file | pkg_mathlib_source_file_chunks | have : ∑ i ∈ Finset.Ioc n m, μ.restrict (disjointed K i) = μ.restrict (K m \ K n) := by
rw [← biUnion_Ioc_disjointed_of_monotone h hm, FiniteMeasure.restrict_biUnion_finset]
· exact (disjoint_disjointed K).set_pairwise _
· exact MeasurableSet.disjointed (fun i ↦ (hK i).measurableSet)
rw [this, res... |
6,942 | lean_file_chunk | 1998c012c7bad7df8a0959cc71b1226fd8ac69a0fd5ea4ca9a4a3a5a0b4d7632 | 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.Map
/-!
# Absolute Continuity of Measures
We say that `μ` is absolutely continuous with respect ... |
17,623 | lean_file_chunk | d75010ea5c48ba1a21f24e82317f2508574638e28d522fd7f81ee5aa248e1831 | 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.Prod
public import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
/-!
# Characterization of a finite measure... |
19,818 | lean_file_chunk | 57c39d05c9da62d5098eabf4957151275a3de01e8f02f6aee91c4bce681a51cb | 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.MeasureTheory.MeasurableSpace.EventuallyMeasurable
public import Mathlib.MeasureTheory.MeasurableSpace.... |
23,991 | lean_file_chunk | c1c15f88234f342449c3199a5745de4b12f009f39527f1e0824a2c0894aeeebf | 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, Peter Pfaffelhuber
-/
module
public import Mathlib.MeasureTheory.SetSemiring
public import Mathlib.MeasureTheory.OuterMeasure.Induced
public import Mathlib.Tactic.FinCases... |
8,910 | lean_file_chunk | 0a5a3caa44768a19a5b84bee2bba924c9abafb3f8c33f70687611b60fac3d1b2 | lean_source_file | pkg_mathlib_source_file_chunks | {S : Finset ι} (hs : ∀ n ∈ S, s n ∈ C) (hS : (S : Set ι).PairwiseDisjoint s) :
m (⋃ i ∈ S, s i) = ∑ i ∈ S, m (s i) := by
classical
induction S using Finset.induction with
| empty => simp
| insert i S hiS h =>
rw [Finset.sum_insert hiS]
simp_rw [← Finset.mem_coe, Finset.coe_insert, Set.biUnion_insert... |
19,038 | lean_file_chunk | 48177261ede215497ed266c811b65fbe413092d1838b36a0c5ef34a4f7a2f99c | 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.MeasureSpace
public import Mathlib.MeasureTheory.Measure.Regular
public import Mathlib.Topology.Sets.Compacts
... |
3,922 | lean_file_chunk | af64f4e0fec9512c382768e297f845849fc33c74ba2eb0e1be4cbefd5955b913 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Integral.Lebesgue.Sub
public import Mathlib.Analysis.Normed.Group.Basic
public impor... |
23,944 | lean_file_chunk | 982b0b2b1ba88a2fd88e178ce726247c2bf8053b72eeeba85e3032161c3eab19 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.Convex.Between
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
public import Mathlib.MeasureTheory.Measure... |
23,974 | lean_file_chunk | 9d60e5877bf737248129307af7db00385052511b7f10cf5c1e265f2f274ff328 | lean_source_file | pkg_mathlib_source_file_chunks | ((hr.eventually (gt_mem_nhds hε)).and (ht.and hst))).exists with
⟨n, hn, hrn, htn, hstn⟩
set u : ℕ → Set X := fun j => ⋃ b ∈ decode₂ (ι n) j, t n b
refine iInf₂_le_of_le u (by rwa [iUnion_decode₂]) ?_
refine iInf_le_of_le (fun j => ?_) ?_
· rw [ediam_iUnion_mem_option]
exact iSup₂_le fun _ _ => (htn _).... |
3,612 | lean_file_chunk | 088e0cd6dbc46695d7cb71d25576bbc8a1289973df03c6f3c7ad0ad22e14e2dd | lean_source_file | pkg_mathlib_source_file_chunks | rw [Set.image_smul, Measure.hausdorffMeasure_smul₀ zero_le_one hn, nnnorm_norm,
NNReal.rpow_one, iso_smul.hausdorffMeasure_image (Or.inl <| zero_le_one' ℝ)]
section NormedFieldAffine
variable [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace P]
variable [MetricSpace P] [NormedAddTorsor... |
23,954 | lean_file_chunk | f0cf6f86c50ef89b68db31f928c91c14a7de2ea7b0cc737fd47cfecd027e4f83 | 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.Countable
public import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
public im... |
12,326 | lean_file_chunk | 9c023db2c7fa3ee150fe59b9fb5dd8069fd8f8733398ee69e24819c0d22462ae | lean_source_file | pkg_mathlib_source_file_chunks | {f : α → ℝ≥0∞} {s : Set α} (hf : AEMeasurable f (μ.restrict s)) (g : α → ℝ≥0∞)
(hs : MeasurableSet s) (h'f : ∀ᵐ x ∂μ.restrict s, f x < ∞) :
∫⁻ a in s, g a ∂μ.withDensity f = ∫⁻ a in s, (f * g) a ∂μ := by
rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul_non_measurable₀ _ hf h'f]
theorem set... |
20,334 | lean_file_chunk | 902da88b88c36213eef8070922a3811f541b30183d24da48fda912c3e2c2b22c | 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.Induced
public import Mathlib.MeasureTheory.OuterMeasure.AE
/-!
# Measure spaces
This file ... |
4,322 | lean_file_chunk | 9ddf208e965a100e08b32db44d25d6277975d4535f5098e0d1ee2b2c94402fc4 | 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.Analysis.Complex.Basic
/-!
# Complex measure
This file defines a complex measure to ... |
11,301 | lean_file_chunk | 9e985dd9fda62b94c0b1c054a501ff2600c307285ca1df86042812bc0642c1c4 | 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.Measure.CharacteristicFunction.Basic
public import Mathlib.MeasureTheory.Measure.Tight
import Mathlib.MeasureTheory.Measure.C... |
20,189 | lean_file_chunk | 94faf3273b4ff4bdb4aefaa0cef63a1589e97fb740ecd2426e10cb80b7abb5db | 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.Trim
public import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
/-!
# Almost everywhere measura... |
15,459 | lean_file_chunk | bdebf78b55339ffc86dcc7ac8a3763a775911639049b50426ad16efc8f7c635b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Group.Measure
public import Mathlib.Tactic.Group
public import Mathlib.Topology.UrysohnsLemma
/-!
# Everywhere posi... |
23,948 | lean_file_chunk | 2f21318bf5fb71d5302ff80edac049c6dee26c7b2a99004eb1ab631229e1d633 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.MeasureTheory.Measure.Portmanteau
public import Mathlib.MeasureTheory.Integral.DominatedConvergence
public import Mathlib.MeasureTheory.Inte... |
14,003 | lean_file_chunk | 24541da88baad1b1c456ab63285ebec306666b44ee7b5cd8d57400c6a1c34138 | lean_source_file | pkg_mathlib_source_file_chunks | · obtain ⟨εs, _, εs_pos, εs_lim⟩ := exists_seq_strictAnti_tendsto (0 : ℝ)
have ε_of_room := Tendsto.add (tendsto_iff_dist_tendsto_zero.mp hμs) εs_lim
have ε_of_room' : Tendsto (fun n ↦ dist (μs n) ν + εs n) atTop (𝓝[>] 0) := by
rw [tendsto_nhdsWithin_iff]
refine ⟨by simpa using ε_of_room, Eventuall... |
10,569 | lean_file_chunk | fae27023939b67284440036efdca14165ce21d0809d492c4cc0990ede3c87f19 | 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.AbsolutelyContinuous
public import Mathlib.MeasureTheory.OuterMeasure.BorelCantelli
/-!
# Quasi-M... |
13,475 | lean_file_chunk | 327b11ba8daaab082d9d0d3f0d1c268015a6691829245ba790c8150950784be0 | 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
-/
module
public import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
/-!
# The Giry monad
Let X be a measurable space. The collection of all measures on X again
... |
5,983 | lean_file_chunk | c62c6853aba9a3487029828afb823df61f8c67a56cc5d9dc3c72ca65982ece77 | 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.Topology.CompactOpen
public import Mathlib.Dynamics.Ergodic.MeasurePreserving
public import Mathlib.MeasureTheory.Measure.Regular
/-!... |
14,323 | lean_file_chunk | 991472be34e62a2872cf2854046358fb102d2aa288d9fad967c289cca114acf4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
public import Mathlib.Topology.MetricSpace.ThickenedIndicator
/-!
# Spaces where indicator... |
12,158 | lean_file_chunk | 51706b5d354e6e238f2e6d8146042023b1dad4be4c542b0bc4d79abae8e3b052 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Oliver Butterley. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Butterley, Yoh Tanimoto
-/
module
public import Mathlib.MeasureTheory.VectorMeasure.Basic
public import Mathlib.Order.Partition.Finpartition
/-!
# Pre-variation of a subaddi... |
23,953 | lean_file_chunk | df1cb571b95b8ca4161339f0e17faaf7c2860eb9f0ce3ecd1620c9f1ec97e544 | 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: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
public import Mathlib.MeasureTheory.Measure.Typeclas... |
11,714 | lean_file_chunk | bc64d40fa9249de8ee9c58ca4eb9810655b1f8eee65c18d5d24e0b0940d8b729 | lean_source_file | pkg_mathlib_source_file_chunks | simp [subsingleton_botSet.eq_singleton_of_mem ha]
obtain ⟨b, hb⟩ : ∃ b, b < a := by simpa only [IsBot, not_forall, not_le] using ha
obtain ⟨u, u_mono, u_lt_a, u_lim⟩ :
∃ u : ℕ → R, StrictMono u ∧ (∀ n : ℕ, u n ∈ Ioo b a) ∧ Tendsto u atTop (𝓝 a) :=
exists_seq_strictMono_tendsto' hb
replace u_lt_a n : u n ... |
5,982 | lean_file_chunk | a3904ee0e31f02faab38a08955022c4b6274378fb68f7f2422d9efc0c317ff60 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich
-/
module
public import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
/-!
# Subtraction of measures
In this file we define `μ - ν` to be the least measure `τ` ... |
10,494 | lean_file_chunk | 0e68a5c8cffb2b9c0bcadd614324b5c78b2c298bb0b705930ea086dd82379d51 | 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.MeasureTheory.Measure.Typeclasses.Finite
public import Mathlib.MeasureTheory.SetSemiring
import Mathlib.Topology.MetricSpace.Lipsc... |
20,974 | lean_file_chunk | 50193b6dc74e2baf38acc5f92fe5624e0a6ffbd15a10a55c0e08d47d88caab2b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
public import Mathlib.MeasureTheory.Measure.Typeclasses.SFinite
/-!
# Measures as r... |
23,960 | lean_file_chunk | 2107fd23926a9f26e7a35b3cca69bf7bf6e9922566863866a3d31b279088489f | 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.Covering.Differentiation
/-!
# Besicovitch covering theorems
The topological Besicovitch covering theorem ensures ... |
23,988 | lean_file_chunk | 8d948044e9c41e36abc7688cb36a0f83d6f5a63d62ab4a5ac0eb9ba9da708ccc | lean_source_file | pkg_mathlib_source_file_chunks | simp only [s, exists_prop, mem_iUnion, mem_ball, mem_singleton_iff, biUnion_and',
exists_eq_left, iUnion_exists, exists_and_left]
exact ⟨⟨p.color a, p.color_lt ha.1 hN⟩, a, rfl, ha⟩
/-!
### The measurable Besicovitch covering theorem
-/
open scoped NNReal
variable [SecondCountableTopology α] [MeasurableSp... |
7,842 | lean_file_chunk | 194dee898bb382cbb61e74693ddbde2828777e4921ad26ac54bb3be9fec04b7f | lean_source_file | pkg_mathlib_source_file_chunks | · -- the sets in the first step have measure at most `μ s + ε / 2`
have A : (∑' x : t0, μ (closedBall x (r x))) ≤ μ s + ε / 2 :=
calc
(∑' x : t0, μ (closedBall x (r x))) = ∑' x : t0, μ (closedBall x (r0 x)) := by
congr 1; ext x; rw [r_t0 x x.2]
_ = μ (⋃ x : t0, closedBall x (r0 x)) :... |
15,087 | lean_file_chunk | 08f0028153db2c308bfd0641b0ceaeb718be3e7743639dc7afd139fc362e569d | 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.Covering.DensityTheorem
/-!
# Liminf, limsup, and uniformly locally doubling measures.
This file is a place to collect lemmas ... |
23,963 | lean_file_chunk | 5fd70236c3f2b75c6eaf3063199e713d504fc40c95dab9987ec9eab5342a2d78 | 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.Constructions.BorelSpace.Basic
public import Mathlib.MeasureTheory.Covering.VitaliFamily
public import Mathlib.Data.... |
1,666 | lean_file_chunk | 7667e838feb293b068e042804e96db9e576a5137d513cacf25e898aedb55cc9b | lean_source_file | pkg_mathlib_source_file_chunks | nonempty_interior _ _ ha := ha.2.1
nontrivial x ε εpos := by
obtain ⟨r, μr, rpos, rε⟩ :
∃ r, μ (closedBall x (3 * r)) ≤ C * μ (closedBall x r) ∧ r ∈ Ioc (0 : ℝ) ε :=
((h x).and_eventually (Ioc_mem_nhdsGT εpos)).exists
refine ⟨closedBall x r, ⟨isClosed_closedBall, ?_, ⟨r, Subset.rfl, μr⟩⟩, by gco... |
23,953 | lean_file_chunk | 3e50f0b9db7b839654537f121a7025db366a9cf7d9a5998d007dcc157e08a2fa | 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.Covering.VitaliFamily
public import Mathlib.MeasureTheory.Function.AEMeasurableOrder
public import Mathlib.MeasureTh... |
23,983 | lean_file_chunk | ded3e684c2f9cba07914df09268ef2d9d8b7f6c01767963ac80dfbaee807eaf5 | lean_source_file | pkg_mathlib_source_file_chunks | have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 hx.2).2 _ (h hx.1)
apply I.frequently.mono fun a ha => ?_
rw [coe_nnreal_smul_apply]
refine (ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le
simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]
/-- If, for al... |
427 | lean_file_chunk | 105cb980e60c0ec3e66402700c0de0746a80f61872d0c962dd215d103821b6ad | lean_source_file | pkg_mathlib_source_file_chunks | filter_upwards [h'x, v.eventually_measure_lt_top x, v.eventually_filterAt_integrableOn x hf]
with a ha h'a h''a
nth_rw 1 [← setAverage_const ha.ne' h'a.ne (f x)]
simp_rw [setAverage_eq']
rw [← integral_sub]
· exact norm_integral_le_integral_norm _
· exact (integrable_inv_smul_measure ha.ne' h'a.ne).2 h''a... |
2,196 | lean_file_chunk | 277131b13276683c1a7f42f450637fdc7acf8b647b2a426fa3ff0f677b28131b | 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.Covering.DensityTheorem
public import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
/-!
# Covering theorems for Leb... |
22,883 | lean_file_chunk | 439b1b8e346a75d2bf42db9e2dee4aa70aa11bd43b345d6f77cb3230b284d67b | 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.Lebesgue.EqHaar
public import Mathlib.MeasureTheory.Covering.Besicovitch
public import Mathlib.Tactic.Adapta... |
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