Measure spaces #
The definition of a measure and a measure space are in MeasureTheory.MeasureSpaceDef, with
only a few basic properties. This file provides many more properties of these objects.
This separation allows the measurability tactic to import only the file MeasureSpaceDef, and to
be available in MeasureSpace (through MeasurableSpace).
Given a measurable space α, a measure on α is a function that sends measurable sets to the
extended nonnegative reals that satisfies the following conditions:
μ ∅ = 0;μis countably additive. This means that the measure of a countable union of pairwise disjoint sets is equal to the measure of the individual sets.
Every measure can be canonically extended to an outer measure, so that it assigns values to all subsets, not just the measurable subsets. On the other hand, a measure that is countably additive on measurable sets can be restricted to measurable sets to obtain a measure. In this file a measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.
Measures on α form a complete lattice, and are closed under scalar multiplication with ℝ≥0∞.
We introduce the following typeclasses for measures:
IsProbabilityMeasure μ:μ univ = 1;IsFiniteMeasure μ:μ univ < ∞;SigmaFinite μ: there exists a countable collection of sets that coverunivwhereμis finite;IsLocallyFiniteMeasure μ:∀ x, ∃ s ∈ 𝓝 x, μ s < ∞;NoAtoms μ:∀ x, μ {x} = 0; possibly should be redefined as∀ s, 0 < μ s → ∃ t ⊆ s, 0 < μ t ∧ μ t < μ s.
Given a measure, the null sets are the sets where μ s = 0, where μ denotes the corresponding
outer measure (so s might not be measurable). We can then define the completion of μ as the
measure on the least σ-algebra that also contains all null sets, by defining the measure to be 0
on the null sets.
Main statements #
completionis the completion of a measure to all null measurable sets.Measure.ofMeasurableandOuterMeasure.toMeasureare two important ways to define a measure.
Implementation notes #
Given μ : Measure α, μ s is the value of the outer measure applied to s.
This conveniently allows us to apply the measure to sets without proving that they are measurable.
We get countable subadditivity for all sets, but only countable additivity for measurable sets.
You often don't want to define a measure via its constructor. Two ways that are sometimes more convenient:
Measure.ofMeasurableis a way to define a measure by only giving its value on measurable sets and proving the properties (1) and (2) mentioned above.OuterMeasure.toMeasureis a way of obtaining a measure from an outer measure by showing that all measurable sets in the measurable space are Carathéodory measurable.
To prove that two measures are equal, there are multiple options:
ext: two measures are equal if they are equal on all measurable sets.ext_of_generateFrom_of_iUnion: two measures are equal if they are equal on a π-system generating the measurable sets, if the π-system contains a spanning increasing sequence of sets where the measures take finite value (in particular the measures are σ-finite). This is a special case of the more generalext_of_generateFrom_of_coverext_of_generate_finite: two finite measures are equal if they are equal on a π-system generating the measurable sets. This is a special case ofext_of_generateFrom_of_iUnionusingC ∪ {univ}, but is easier to work with.
A MeasureSpace is a class that is a measurable space with a canonical measure.
The measure is denoted volume.
References #
- https://en.wikipedia.org/wiki/Measure_(mathematics)
- https://en.wikipedia.org/wiki/Complete_measure
- https://en.wikipedia.org/wiki/Almost_everywhere
Tags #
measure, almost everywhere, measure space, completion, null set, null measurable set
instance
MeasureTheory.ae_isMeasurablyGenerated
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
theorem
MeasureTheory.ae_uIoc_iff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[LinearOrder α]
{a : α}
{b : α}
{P : α → Prop}
:
See also MeasureTheory.ae_restrict_uIoc_iff.
theorem
MeasureTheory.measure_union
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s₁ : Set α}
{s₂ : Set α}
(hd : Disjoint s₁ s₂)
(h : MeasurableSet s₂)
:
theorem
MeasureTheory.measure_union'
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s₁ : Set α}
{s₂ : Set α}
(hd : Disjoint s₁ s₂)
(h : MeasurableSet s₁)
:
theorem
MeasureTheory.measure_inter_add_diff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{t : Set α}
(s : Set α)
(ht : MeasurableSet t)
:
theorem
MeasureTheory.measure_diff_add_inter
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{t : Set α}
(s : Set α)
(ht : MeasurableSet t)
:
theorem
MeasureTheory.measure_union_add_inter
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{t : Set α}
(s : Set α)
(ht : MeasurableSet t)
:
theorem
MeasureTheory.measure_union_add_inter'
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(hs : MeasurableSet s)
(t : Set α)
:
theorem
MeasureTheory.measure_symmDiff_eq
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
(ht : MeasurableSet t)
:
theorem
MeasureTheory.measure_symmDiff_le
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
(s : Set α)
(t : Set α)
(u : Set α)
:
theorem
MeasureTheory.measure_add_measure_compl
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(h : MeasurableSet s)
:
theorem
MeasureTheory.measure_biUnion₀
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set β}
{f : β → Set α}
(hs : Set.Countable s)
(hd : Set.Pairwise s (MeasureTheory.AEDisjoint μ on f))
(h : ∀ (b : β), b ∈ s → MeasureTheory.NullMeasurableSet (f b))
:
theorem
MeasureTheory.measure_biUnion
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set β}
{f : β → Set α}
(hs : Set.Countable s)
(hd : Set.PairwiseDisjoint s f)
(h : ∀ (b : β), b ∈ s → MeasurableSet (f b))
:
theorem
MeasureTheory.measure_sUnion₀
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{S : Set (Set α)}
(hs : Set.Countable S)
(hd : Set.Pairwise S (MeasureTheory.AEDisjoint μ))
(h : ∀ (s : Set α), s ∈ S → MeasureTheory.NullMeasurableSet s)
:
theorem
MeasureTheory.measure_sUnion
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{S : Set (Set α)}
(hs : Set.Countable S)
(hd : Set.Pairwise S Disjoint)
(h : ∀ (s : Set α), s ∈ S → MeasurableSet s)
:
theorem
MeasureTheory.measure_biUnion_finset₀
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Finset ι}
{f : ι → Set α}
(hd : Set.Pairwise (↑s) (MeasureTheory.AEDisjoint μ on f))
(hm : ∀ (b : ι), b ∈ s → MeasureTheory.NullMeasurableSet (f b))
:
↑↑μ (⋃ (b : ι) (_ : b ∈ s), f b) = Finset.sum s fun p => ↑↑μ (f p)
theorem
MeasureTheory.measure_biUnion_finset
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Finset ι}
{f : ι → Set α}
(hd : Set.PairwiseDisjoint (↑s) f)
(hm : ∀ (b : ι), b ∈ s → MeasurableSet (f b))
:
↑↑μ (⋃ (b : ι) (_ : b ∈ s), f b) = Finset.sum s fun p => ↑↑μ (f p)
theorem
MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint₀
{α : Type u_1}
{ι : Type u_8}
[MeasurableSpace α]
(μ : MeasureTheory.Measure α)
{As : ι → Set α}
(As_mble : ∀ (i : ι), MeasureTheory.NullMeasurableSet (As i))
(As_disj : Pairwise (MeasureTheory.AEDisjoint μ on As))
:
∑' (i : ι), ↑↑μ (As i) ≤ ↑↑μ (⋃ (i : ι), As i)
The measure of an a.e. disjoint union (even uncountable) of null-measurable sets is at least the sum of the measures of the sets.
theorem
MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint
{α : Type u_1}
{ι : Type u_8}
[MeasurableSpace α]
(μ : MeasureTheory.Measure α)
{As : ι → Set α}
(As_mble : ∀ (i : ι), MeasurableSet (As i))
(As_disj : Pairwise (Disjoint on As))
:
∑' (i : ι), ↑↑μ (As i) ≤ ↑↑μ (⋃ (i : ι), As i)
The measure of a disjoint union (even uncountable) of measurable sets is at least the sum of the measures of the sets.
theorem
MeasureTheory.tsum_measure_preimage_singleton
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set β}
(hs : Set.Countable s)
{f : α → β}
(hf : ∀ (y : β), y ∈ s → MeasurableSet (f ⁻¹' {y}))
:
If s is a countable set, then the measure of its preimage can be found as the sum of measures
of the fibers f ⁻¹' {y}.
theorem
MeasureTheory.sum_measure_preimage_singleton
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
(s : Finset β)
{f : α → β}
(hf : ∀ (y : β), y ∈ s → MeasurableSet (f ⁻¹' {y}))
:
(Finset.sum s fun b => ↑↑μ (f ⁻¹' {b})) = ↑↑μ (f ⁻¹' ↑s)
If s is a Finset, then the measure of its preimage can be found as the sum of measures
of the fibers f ⁻¹' {y}.
theorem
MeasureTheory.measure_diff_null'
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s₁ : Set α}
{s₂ : Set α}
(h : ↑↑μ (s₁ ∩ s₂) = 0)
:
theorem
MeasureTheory.measure_diff_null
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s₁ : Set α}
{s₂ : Set α}
(h : ↑↑μ s₂ = 0)
:
theorem
MeasureTheory.measure_add_diff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(hs : MeasurableSet s)
(t : Set α)
:
theorem
MeasureTheory.measure_diff'
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{t : Set α}
(s : Set α)
(hm : MeasurableSet t)
(h_fin : ↑↑μ t ≠ ⊤)
:
theorem
MeasureTheory.measure_diff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s₁ : Set α}
{s₂ : Set α}
(h : s₂ ⊆ s₁)
(h₂ : MeasurableSet s₂)
(h_fin : ↑↑μ s₂ ≠ ⊤)
:
theorem
MeasureTheory.le_measure_diff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s₁ : Set α}
{s₂ : Set α}
:
theorem
MeasureTheory.measure_diff_lt_of_lt_add
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
(hst : s ⊆ t)
(hs' : ↑↑μ s ≠ ⊤)
{ε : ENNReal}
(h : ↑↑μ t < ↑↑μ s + ε)
:
theorem
MeasureTheory.measure_diff_le_iff_le_add
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
(hst : s ⊆ t)
(hs' : ↑↑μ s ≠ ⊤)
{ε : ENNReal}
:
theorem
MeasureTheory.measure_eq_measure_of_null_diff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hst : s ⊆ t)
(h_nulldiff : ↑↑μ (t \ s) = 0)
:
↑↑μ s = ↑↑μ t
theorem
MeasureTheory.measure_eq_measure_of_between_null_diff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s₁ : Set α}
{s₂ : Set α}
{s₃ : Set α}
(h12 : s₁ ⊆ s₂)
(h23 : s₂ ⊆ s₃)
(h_nulldiff : ↑↑μ (s₃ \ s₁) = 0)
:
theorem
MeasureTheory.measure_eq_measure_smaller_of_between_null_diff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s₁ : Set α}
{s₂ : Set α}
{s₃ : Set α}
(h12 : s₁ ⊆ s₂)
(h23 : s₂ ⊆ s₃)
(h_nulldiff : ↑↑μ (s₃ \ s₁) = 0)
:
↑↑μ s₁ = ↑↑μ s₂
theorem
MeasureTheory.measure_eq_measure_larger_of_between_null_diff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s₁ : Set α}
{s₂ : Set α}
{s₃ : Set α}
(h12 : s₁ ⊆ s₂)
(h23 : s₂ ⊆ s₃)
(h_nulldiff : ↑↑μ (s₃ \ s₁) = 0)
:
↑↑μ s₂ = ↑↑μ s₃
theorem
MeasureTheory.measure_compl
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(h₁ : MeasurableSet s)
(h_fin : ↑↑μ s ≠ ⊤)
:
@[simp]
theorem
MeasureTheory.union_ae_eq_left_iff_ae_subset
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
:
s ∪ t =ᶠ[MeasureTheory.Measure.ae μ] s ↔ t ≤ᶠ[MeasureTheory.Measure.ae μ] s
@[simp]
theorem
MeasureTheory.union_ae_eq_right_iff_ae_subset
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
:
s ∪ t =ᶠ[MeasureTheory.Measure.ae μ] t ↔ s ≤ᶠ[MeasureTheory.Measure.ae μ] t
theorem
MeasureTheory.ae_eq_of_ae_subset_of_measure_ge
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h₁ : s ≤ᶠ[MeasureTheory.Measure.ae μ] t)
(h₂ : ↑↑μ t ≤ ↑↑μ s)
(hsm : MeasurableSet s)
(ht : ↑↑μ t ≠ ⊤)
:
theorem
MeasureTheory.ae_eq_of_subset_of_measure_ge
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h₁ : s ⊆ t)
(h₂ : ↑↑μ t ≤ ↑↑μ s)
(hsm : MeasurableSet s)
(ht : ↑↑μ t ≠ ⊤)
:
If s ⊆ t, μ t ≤ μ s, μ t ≠ ∞, and s is measurable, then s =ᵐ[μ] t.
theorem
MeasureTheory.measure_iUnion_congr_of_subset
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable β]
{s : β → Set α}
{t : β → Set α}
(hsub : ∀ (b : β), s b ⊆ t b)
(h_le : ∀ (b : β), ↑↑μ (t b) ≤ ↑↑μ (s b))
:
↑↑μ (⋃ (b : β), s b) = ↑↑μ (⋃ (b : β), t b)
@[simp]
theorem
MeasureTheory.measure_iUnion_toMeasurable
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable β]
(s : β → Set α)
:
↑↑μ (⋃ (b : β), MeasureTheory.toMeasurable μ (s b)) = ↑↑μ (⋃ (b : β), s b)
theorem
MeasureTheory.measure_biUnion_toMeasurable
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{I : Set β}
(hc : Set.Countable I)
(s : β → Set α)
:
↑↑μ (⋃ (b : β) (_ : b ∈ I), MeasureTheory.toMeasurable μ (s b)) = ↑↑μ (⋃ (b : β) (_ : b ∈ I), s b)
@[simp]
theorem
MeasureTheory.measure_toMeasurable_union
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
:
↑↑μ (MeasureTheory.toMeasurable μ s ∪ t) = ↑↑μ (s ∪ t)
@[simp]
theorem
MeasureTheory.measure_union_toMeasurable
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
:
↑↑μ (s ∪ MeasureTheory.toMeasurable μ t) = ↑↑μ (s ∪ t)
theorem
MeasureTheory.sum_measure_le_measure_univ
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Finset ι}
{t : ι → Set α}
(h : ∀ (i : ι), i ∈ s → MeasurableSet (t i))
(H : Set.PairwiseDisjoint (↑s) t)
:
(Finset.sum s fun i => ↑↑μ (t i)) ≤ ↑↑μ Set.univ
theorem
MeasureTheory.tsum_measure_le_measure_univ
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : ι → Set α}
(hs : ∀ (i : ι), MeasurableSet (s i))
(H : Pairwise (Disjoint on s))
:
∑' (i : ι), ↑↑μ (s i) ≤ ↑↑μ Set.univ
theorem
MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
{s : ι → Set α}
(hs : ∀ (i : ι), MeasurableSet (s i))
(H : ↑↑μ Set.univ < ∑' (i : ι), ↑↑μ (s i))
:
∃ i j _h, Set.Nonempty (s i ∩ s j)
Pigeonhole principle for measure spaces: if ∑' i, μ (s i) > μ univ, then
one of the intersections s i ∩ s j is not empty.
theorem
MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_sum_measure
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
{s : Finset ι}
{t : ι → Set α}
(h : ∀ (i : ι), i ∈ s → MeasurableSet (t i))
(H : ↑↑μ Set.univ < Finset.sum s fun i => ↑↑μ (t i))
:
Pigeonhole principle for measure spaces: if s is a Finset and
∑ i in s, μ (t i) > μ univ, then one of the intersections t i ∩ t j is not empty.
theorem
MeasureTheory.nonempty_inter_of_measure_lt_add
{α : Type u_1}
{m : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
{s : Set α}
{t : Set α}
{u : Set α}
(ht : MeasurableSet t)
(h's : s ⊆ u)
(h't : t ⊆ u)
(h : ↑↑μ u < ↑↑μ s + ↑↑μ t)
:
Set.Nonempty (s ∩ t)
If two sets s and t are included in a set u, and μ s + μ t > μ u,
then s intersects t. Version assuming that t is measurable.
theorem
MeasureTheory.nonempty_inter_of_measure_lt_add'
{α : Type u_1}
{m : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
{s : Set α}
{t : Set α}
{u : Set α}
(hs : MeasurableSet s)
(h's : s ⊆ u)
(h't : t ⊆ u)
(h : ↑↑μ u < ↑↑μ s + ↑↑μ t)
:
Set.Nonempty (s ∩ t)
If two sets s and t are included in a set u, and μ s + μ t > μ u,
then s intersects t. Version assuming that s is measurable.
theorem
MeasureTheory.measure_iUnion_eq_iSup
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
(hd : Directed (fun x x_1 => x ⊆ x_1) s)
:
↑↑μ (⋃ (i : ι), s i) = ⨆ (i : ι), ↑↑μ (s i)
Continuity from below: the measure of the union of a directed sequence of (not necessarily -measurable) sets is the supremum of the measures.
theorem
MeasureTheory.measure_biUnion_eq_iSup
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : ι → Set α}
{t : Set ι}
(ht : Set.Countable t)
(hd : DirectedOn ((fun x x_1 => x ⊆ x_1) on s) t)
:
theorem
MeasureTheory.measure_iInter_eq_iInf
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
(h : ∀ (i : ι), MeasurableSet (s i))
(hd : Directed (fun x x_1 => x ⊇ x_1) s)
(hfin : ∃ i, ↑↑μ (s i) ≠ ⊤)
:
↑↑μ (⋂ (i : ι), s i) = ⨅ (i : ι), ↑↑μ (s i)
Continuity from above: the measure of the intersection of a decreasing sequence of measurable sets is the infimum of the measures.
theorem
MeasureTheory.tendsto_measure_iUnion
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[SemilatticeSup ι]
[Countable ι]
{s : ι → Set α}
(hm : Monotone s)
:
Filter.Tendsto (↑↑μ ∘ s) Filter.atTop (nhds (↑↑μ (⋃ (n : ι), s n)))
Continuity from below: the measure of the union of an increasing sequence of measurable sets is the limit of the measures.
theorem
MeasureTheory.tendsto_measure_iInter
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable ι]
[SemilatticeSup ι]
{s : ι → Set α}
(hs : ∀ (n : ι), MeasurableSet (s n))
(hm : Antitone s)
(hf : ∃ i, ↑↑μ (s i) ≠ ⊤)
:
Filter.Tendsto (↑↑μ ∘ s) Filter.atTop (nhds (↑↑μ (⋂ (n : ι), s n)))
Continuity from above: the measure of the intersection of a decreasing sequence of measurable sets is the limit of the measures.
theorem
MeasureTheory.tendsto_measure_biInter_gt
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ι : Type u_8}
[LinearOrder ι]
[TopologicalSpace ι]
[OrderTopology ι]
[DenselyOrdered ι]
[TopologicalSpace.FirstCountableTopology ι]
{s : ι → Set α}
{a : ι}
(hs : ∀ (r : ι), r > a → MeasurableSet (s r))
(hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j)
(hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤)
:
Filter.Tendsto (↑↑μ ∘ s) (nhdsWithin a (Set.Ioi a)) (nhds (↑↑μ (⋂ (r : ι) (_ : r > a), s r)))
The measure of the intersection of a decreasing sequence of measurable sets indexed by a linear order with first countable topology is the limit of the measures.
theorem
MeasureTheory.measure_limsup_eq_zero
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : ℕ → Set α}
(hs : ∑' (i : ℕ), ↑↑μ (s i) ≠ ⊤)
:
↑↑μ (Filter.limsup s Filter.atTop) = 0
One direction of the Borel-Cantelli lemma: if (sᵢ) is a sequence of sets such
that ∑ μ sᵢ is finite, then the limit superior of the sᵢ is a null set.
theorem
MeasureTheory.measure_liminf_eq_zero
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : ℕ → Set α}
(h : ∑' (i : ℕ), ↑↑μ (s i) ≠ ⊤)
:
↑↑μ (Filter.liminf s Filter.atTop) = 0
theorem
MeasureTheory.limsup_ae_eq_of_forall_ae_eq
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
(s : ℕ → Set α)
{t : Set α}
(h : ∀ (n : ℕ), s n =ᶠ[MeasureTheory.Measure.ae μ] t)
:
Filter.limsup s Filter.atTop =ᶠ[MeasureTheory.Measure.ae μ] t
theorem
MeasureTheory.liminf_ae_eq_of_forall_ae_eq
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
(s : ℕ → Set α)
{t : Set α}
(h : ∀ (n : ℕ), s n =ᶠ[MeasureTheory.Measure.ae μ] t)
:
Filter.liminf s Filter.atTop =ᶠ[MeasureTheory.Measure.ae μ] t
theorem
MeasureTheory.measure_if
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{x : β}
{t : Set β}
{s : Set α}
:
↑↑μ (if x ∈ t then s else ∅) = Set.indicator t (fun x => ↑↑μ s) x
def
MeasureTheory.OuterMeasure.toMeasure
{α : Type u_1}
[ms : MeasurableSpace α]
(m : MeasureTheory.OuterMeasure α)
(h : ms ≤ MeasureTheory.OuterMeasure.caratheodory m)
:
Obtain a measure by giving an outer measure where all sets in the σ-algebra are Carathéodory measurable.
Instances For
theorem
MeasureTheory.le_toOuterMeasure_caratheodory
{α : Type u_1}
[ms : MeasurableSpace α]
(μ : MeasureTheory.Measure α)
:
@[simp]
theorem
MeasureTheory.toMeasure_toOuterMeasure
{α : Type u_1}
[ms : MeasurableSpace α]
(m : MeasureTheory.OuterMeasure α)
(h : ms ≤ MeasureTheory.OuterMeasure.caratheodory m)
:
theorem
MeasureTheory.toMeasure_apply
{α : Type u_1}
[ms : MeasurableSpace α]
(m : MeasureTheory.OuterMeasure α)
(h : ms ≤ MeasureTheory.OuterMeasure.caratheodory m)
{s : Set α}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.OuterMeasure.toMeasure m h) s = ↑m s
theorem
MeasureTheory.le_toMeasure_apply
{α : Type u_1}
[ms : MeasurableSpace α]
(m : MeasureTheory.OuterMeasure α)
(h : ms ≤ MeasureTheory.OuterMeasure.caratheodory m)
(s : Set α)
:
↑m s ≤ ↑↑(MeasureTheory.OuterMeasure.toMeasure m h) s
theorem
MeasureTheory.toMeasure_apply₀
{α : Type u_1}
[ms : MeasurableSpace α]
(m : MeasureTheory.OuterMeasure α)
(h : ms ≤ MeasureTheory.OuterMeasure.caratheodory m)
{s : Set α}
(hs : MeasureTheory.NullMeasurableSet s)
:
↑↑(MeasureTheory.OuterMeasure.toMeasure m h) s = ↑m s
@[simp]
theorem
MeasureTheory.toOuterMeasure_toMeasure
{α : Type u_1}
[ms : MeasurableSpace α]
{μ : MeasureTheory.Measure α}
:
MeasureTheory.OuterMeasure.toMeasure ↑μ (_ : ms ≤ MeasureTheory.OuterMeasure.caratheodory ↑μ) = μ
theorem
MeasureTheory.boundedBy_measure
{α : Type u_1}
[ms : MeasurableSpace α]
(μ : MeasureTheory.Measure α)
:
theorem
MeasureTheory.Measure.measure_inter_eq_of_measure_eq
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
{u : Set α}
(hs : MeasurableSet s)
(h : ↑↑μ t = ↑↑μ u)
(htu : t ⊆ u)
(ht_ne_top : ↑↑μ t ≠ ⊤)
:
If u is a superset of t with the same (finite) measure (both sets possibly non-measurable),
then for any measurable set s one also has μ (t ∩ s) = μ (u ∩ s).
theorem
MeasureTheory.Measure.measure_toMeasurable_inter
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
(ht : ↑↑μ t ≠ ⊤)
:
↑↑μ (MeasureTheory.toMeasurable μ t ∩ s) = ↑↑μ (t ∩ s)
The measurable superset toMeasurable μ t of t (which has the same measure as t)
satisfies, for any measurable set s, the equality μ (toMeasurable μ t ∩ s) = μ (u ∩ s).
Here, we require that the measure of t is finite. The conclusion holds without this assumption
when the measure is sigma_finite, see measure_toMeasurable_inter_of_sigmaFinite.
The ℝ≥0∞-module of measures #
@[simp]
@[simp]
instance
MeasureTheory.Measure.instSubsingleton
{α : Type u_1}
[IsEmpty α]
{m : MeasurableSpace α}
:
theorem
MeasureTheory.Measure.eq_zero_of_isEmpty
{α : Type u_1}
[IsEmpty α]
{_m : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
:
μ = 0
@[simp]
theorem
MeasureTheory.Measure.add_toOuterMeasure
{α : Type u_1}
{_m : MeasurableSpace α}
(μ₁ : MeasureTheory.Measure α)
(μ₂ : MeasureTheory.Measure α)
:
@[simp]
theorem
MeasureTheory.Measure.coe_add
{α : Type u_1}
{_m : MeasurableSpace α}
(μ₁ : MeasureTheory.Measure α)
(μ₂ : MeasureTheory.Measure α)
:
theorem
MeasureTheory.Measure.add_apply
{α : Type u_1}
{_m : MeasurableSpace α}
(μ₁ : MeasureTheory.Measure α)
(μ₂ : MeasureTheory.Measure α)
(s : Set α)
:
instance
MeasureTheory.Measure.instSMul
{α : Type u_1}
{R : Type u_6}
[SMul R ENNReal]
[IsScalarTower R ENNReal ENNReal]
[MeasurableSpace α]
:
SMul R (MeasureTheory.Measure α)
@[simp]
theorem
MeasureTheory.Measure.smul_toOuterMeasure
{α : Type u_1}
{R : Type u_6}
[SMul R ENNReal]
[IsScalarTower R ENNReal ENNReal]
{_m : MeasurableSpace α}
(c : R)
(μ : MeasureTheory.Measure α)
:
@[simp]
theorem
MeasureTheory.Measure.coe_smul
{α : Type u_1}
{R : Type u_6}
[SMul R ENNReal]
[IsScalarTower R ENNReal ENNReal]
{_m : MeasurableSpace α}
(c : R)
(μ : MeasureTheory.Measure α)
:
@[simp]
theorem
MeasureTheory.Measure.smul_apply
{α : Type u_1}
{R : Type u_6}
[SMul R ENNReal]
[IsScalarTower R ENNReal ENNReal]
{_m : MeasurableSpace α}
(c : R)
(μ : MeasureTheory.Measure α)
(s : Set α)
:
instance
MeasureTheory.Measure.instSMulCommClass
{α : Type u_1}
{R : Type u_6}
{R' : Type u_7}
[SMul R ENNReal]
[IsScalarTower R ENNReal ENNReal]
[SMul R' ENNReal]
[IsScalarTower R' ENNReal ENNReal]
[SMulCommClass R R' ENNReal]
[MeasurableSpace α]
:
SMulCommClass R R' (MeasureTheory.Measure α)
instance
MeasureTheory.Measure.instIsScalarTower
{α : Type u_1}
{R : Type u_6}
{R' : Type u_7}
[SMul R ENNReal]
[IsScalarTower R ENNReal ENNReal]
[SMul R' ENNReal]
[IsScalarTower R' ENNReal ENNReal]
[SMul R R']
[IsScalarTower R R' ENNReal]
[MeasurableSpace α]
:
IsScalarTower R R' (MeasureTheory.Measure α)
instance
MeasureTheory.Measure.instIsCentralScalar
{α : Type u_1}
{R : Type u_6}
[SMul R ENNReal]
[IsScalarTower R ENNReal ENNReal]
[SMul Rᵐᵒᵖ ENNReal]
[IsCentralScalar R ENNReal]
[MeasurableSpace α]
:
instance
MeasureTheory.Measure.instMulAction
{α : Type u_1}
{R : Type u_6}
[Monoid R]
[MulAction R ENNReal]
[IsScalarTower R ENNReal ENNReal]
[MeasurableSpace α]
:
def
MeasureTheory.Measure.coeAddHom
{α : Type u_1}
:
{x : MeasurableSpace α} → MeasureTheory.Measure α →+ Set α → ENNReal
Coercion to function as an additive monoid homomorphism.
Instances For
@[simp]
theorem
MeasureTheory.Measure.coe_finset_sum
{α : Type u_1}
{ι : Type u_5}
{_m : MeasurableSpace α}
(I : Finset ι)
(μ : ι → MeasureTheory.Measure α)
:
↑↑(Finset.sum I fun i => μ i) = Finset.sum I fun i => ↑↑(μ i)
theorem
MeasureTheory.Measure.finset_sum_apply
{α : Type u_1}
{ι : Type u_5}
{m : MeasurableSpace α}
(I : Finset ι)
(μ : ι → MeasureTheory.Measure α)
(s : Set α)
:
↑↑(Finset.sum I fun i => μ i) s = Finset.sum I fun i => ↑↑(μ i) s
instance
MeasureTheory.Measure.instDistribMulAction
{α : Type u_1}
{R : Type u_6}
[Monoid R]
[DistribMulAction R ENNReal]
[IsScalarTower R ENNReal ENNReal]
[MeasurableSpace α]
:
instance
MeasureTheory.Measure.instModule
{α : Type u_1}
{R : Type u_6}
[Semiring R]
[Module R ENNReal]
[IsScalarTower R ENNReal ENNReal]
[MeasurableSpace α]
:
Module R (MeasureTheory.Measure α)
theorem
MeasureTheory.Measure.coe_nnreal_smul_apply
{α : Type u_1}
{_m : MeasurableSpace α}
(c : NNReal)
(μ : MeasureTheory.Measure α)
(s : Set α)
:
@[simp]
theorem
MeasureTheory.Measure.nnreal_smul_coe_apply
{α : Type u_1}
{_m : MeasurableSpace α}
(c : NNReal)
(μ : MeasureTheory.Measure α)
(s : Set α)
:
theorem
MeasureTheory.Measure.ae_smul_measure_iff
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{p : α → Prop}
{c : ENNReal}
(hc : c ≠ 0)
:
theorem
MeasureTheory.Measure.measure_eq_left_of_subset_of_measure_add_eq
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : ↑↑(μ + ν) t ≠ ⊤)
(h' : s ⊆ t)
(h'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t)
:
↑↑μ s = ↑↑μ t
theorem
MeasureTheory.Measure.measure_eq_right_of_subset_of_measure_add_eq
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : ↑↑(μ + ν) t ≠ ⊤)
(h' : s ⊆ t)
(h'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t)
:
↑↑ν s = ↑↑ν t
theorem
MeasureTheory.Measure.measure_toMeasurable_add_inter_left
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
(ht : ↑↑(μ + ν) t ≠ ⊤)
:
↑↑μ (MeasureTheory.toMeasurable (μ + ν) t ∩ s) = ↑↑μ (t ∩ s)
theorem
MeasureTheory.Measure.measure_toMeasurable_add_inter_right
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
(ht : ↑↑(μ + ν) t ≠ ⊤)
:
↑↑ν (MeasureTheory.toMeasurable (μ + ν) t ∩ s) = ↑↑ν (t ∩ s)
The complete lattice of measures #
Measures are partially ordered.
The definition of less equal here is equivalent to the definition without the
measurable set condition, and this is shown by Measure.le_iff'. It is defined
this way since, to prove μ ≤ ν, we may simply intros s hs instead of rewriting followed
by intros s hs.
theorem
MeasureTheory.Measure.le_iff
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ₁ : MeasureTheory.Measure α}
{μ₂ : MeasureTheory.Measure α}
:
μ₁ ≤ μ₂ ↔ ∀ (s : Set α), MeasurableSet s → ↑↑μ₁ s ≤ ↑↑μ₂ s
theorem
MeasureTheory.Measure.toOuterMeasure_le
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ₁ : MeasureTheory.Measure α}
{μ₂ : MeasureTheory.Measure α}
:
theorem
MeasureTheory.Measure.le_iff'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ₁ : MeasureTheory.Measure α}
{μ₂ : MeasureTheory.Measure α}
:
theorem
MeasureTheory.Measure.lt_iff
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
:
theorem
MeasureTheory.Measure.lt_iff'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
:
instance
MeasureTheory.Measure.covariantAddLE
{α : Type u_1}
[MeasurableSpace α]
:
CovariantClass (MeasureTheory.Measure α) (MeasureTheory.Measure α) (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
theorem
MeasureTheory.Measure.le_add_left
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{ν' : MeasureTheory.Measure α}
(h : μ ≤ ν)
:
theorem
MeasureTheory.Measure.le_add_right
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{ν' : MeasureTheory.Measure α}
(h : μ ≤ ν)
:
theorem
MeasureTheory.Measure.sInf_caratheodory
{α : Type u_1}
{m0 : MeasurableSpace α}
{m : Set (MeasureTheory.Measure α)}
(s : Set α)
(hs : MeasurableSet s)
:
theorem
MeasureTheory.Measure.sInf_apply
{α : Type u_1}
{m0 : MeasurableSpace α}
{s : Set α}
{m : Set (MeasureTheory.Measure α)}
(hs : MeasurableSet s)
:
@[simp]
@[simp]
@[simp]
theorem
MeasureTheory.Measure.top_add
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
@[simp]
theorem
MeasureTheory.Measure.add_top
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
theorem
MeasureTheory.Measure.zero_le
{α : Type u_1}
{_m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
:
0 ≤ μ
theorem
MeasureTheory.Measure.nonpos_iff_eq_zero'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
@[simp]
theorem
MeasureTheory.Measure.measure_univ_eq_zero
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
theorem
MeasureTheory.Measure.measure_univ_ne_zero
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
instance
MeasureTheory.Measure.instNeZeroENNRealInstENNRealZeroMeasureOfToOuterMeasureUniv
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NeZero μ]
:
NeZero (↑↑μ Set.univ)
@[simp]
theorem
MeasureTheory.Measure.measure_univ_pos
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
Pushforward and pullback #
def
MeasureTheory.Measure.liftLinear
{α : Type u_1}
{β : Type u_2}
[MeasurableSpace β]
{m0 : MeasurableSpace α}
(f : MeasureTheory.OuterMeasure α →ₗ[ENNReal] MeasureTheory.OuterMeasure β)
(hf : ∀ (μ : MeasureTheory.Measure α), inst✝ ≤ MeasureTheory.OuterMeasure.caratheodory (↑f ↑μ))
:
Lift a linear map between OuterMeasure spaces such that for each measure μ every measurable
set is caratheodory-measurable w.r.t. f μ to a linear map between Measure spaces.
Instances For
@[simp]
theorem
MeasureTheory.Measure.liftLinear_apply
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : MeasureTheory.OuterMeasure α →ₗ[ENNReal] MeasureTheory.OuterMeasure β}
(hf : ∀ (μ : MeasureTheory.Measure α), inst✝ ≤ MeasureTheory.OuterMeasure.caratheodory (↑f ↑μ))
{s : Set β}
(hs : MeasurableSet s)
:
↑↑(↑(MeasureTheory.Measure.liftLinear f hf) μ) s = ↑(↑f ↑μ) s
theorem
MeasureTheory.Measure.le_liftLinear_apply
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : MeasureTheory.OuterMeasure α →ₗ[ENNReal] MeasureTheory.OuterMeasure β}
(hf : ∀ (μ : MeasureTheory.Measure α), inst✝ ≤ MeasureTheory.OuterMeasure.caratheodory (↑f ↑μ))
(s : Set β)
:
↑(↑f ↑μ) s ≤ ↑↑(↑(MeasureTheory.Measure.liftLinear f hf) μ) s
def
MeasureTheory.Measure.mapₗ
{α : Type u_1}
{β : Type u_2}
[MeasurableSpace β]
[MeasurableSpace α]
(f : α → β)
:
The pushforward of a measure as a linear map. It is defined to be 0 if f is not
a measurable function.
Instances For
theorem
MeasureTheory.Measure.mapₗ_congr
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
{g : α → β}
(hf : Measurable f)
(hg : Measurable g)
(h : f =ᶠ[MeasureTheory.Measure.ae μ] g)
:
↑(MeasureTheory.Measure.mapₗ f) μ = ↑(MeasureTheory.Measure.mapₗ g) μ
theorem
MeasureTheory.Measure.map_def
{α : Type u_8}
{β : Type u_9}
[MeasurableSpace β]
[MeasurableSpace α]
(f : α → β)
(μ : MeasureTheory.Measure α)
:
MeasureTheory.Measure.map f μ = if hf : AEMeasurable f then ↑(MeasureTheory.Measure.mapₗ (AEMeasurable.mk f hf)) μ else 0
@[irreducible]
def
MeasureTheory.Measure.map
{α : Type u_8}
{β : Type u_9}
[MeasurableSpace β]
[MeasurableSpace α]
(f : α → β)
(μ : MeasureTheory.Measure α)
:
The pushforward of a measure. It is defined to be 0 if f is not an almost everywhere
measurable function.
Instances For
theorem
MeasureTheory.Measure.mapₗ_mk_apply_of_aemeasurable
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
(hf : AEMeasurable f)
:
↑(MeasureTheory.Measure.mapₗ (AEMeasurable.mk f hf)) μ = MeasureTheory.Measure.map f μ
theorem
MeasureTheory.Measure.mapₗ_apply_of_measurable
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{f : α → β}
(hf : Measurable f)
(μ : MeasureTheory.Measure α)
:
↑(MeasureTheory.Measure.mapₗ f) μ = MeasureTheory.Measure.map f μ
@[simp]
theorem
MeasureTheory.Measure.map_add
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
(μ : MeasureTheory.Measure α)
(ν : MeasureTheory.Measure α)
{f : α → β}
(hf : Measurable f)
:
MeasureTheory.Measure.map f (μ + ν) = MeasureTheory.Measure.map f μ + MeasureTheory.Measure.map f ν
@[simp]
theorem
MeasureTheory.Measure.map_zero
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
(f : α → β)
:
MeasureTheory.Measure.map f 0 = 0
theorem
MeasureTheory.Measure.map_of_not_aemeasurable
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{f : α → β}
{μ : MeasureTheory.Measure α}
(hf : ¬AEMeasurable f)
:
MeasureTheory.Measure.map f μ = 0
theorem
MeasureTheory.Measure.map_congr
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
{g : α → β}
(h : f =ᶠ[MeasureTheory.Measure.ae μ] g)
:
@[simp]
theorem
MeasureTheory.Measure.map_smul
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
(c : ENNReal)
(μ : MeasureTheory.Measure α)
(f : α → β)
:
MeasureTheory.Measure.map f (c • μ) = c • MeasureTheory.Measure.map f μ
@[simp]
theorem
MeasureTheory.Measure.map_smul_nnreal
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
(c : NNReal)
(μ : MeasureTheory.Measure α)
(f : α → β)
:
MeasureTheory.Measure.map f (c • μ) = c • MeasureTheory.Measure.map f μ
@[simp]
theorem
MeasureTheory.Measure.map_apply_of_aemeasurable
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
(hf : AEMeasurable f)
{s : Set β}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.map f μ) s = ↑↑μ (f ⁻¹' s)
We can evaluate the pushforward on measurable sets. For non-measurable sets, see
MeasureTheory.Measure.le_map_apply and MeasurableEquiv.map_apply.
@[simp]
theorem
MeasureTheory.Measure.map_apply
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
(hf : Measurable f)
{s : Set β}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.map f μ) s = ↑↑μ (f ⁻¹' s)
theorem
MeasureTheory.Measure.map_toOuterMeasure
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
(hf : AEMeasurable f)
:
@[simp]
theorem
MeasureTheory.Measure.map_id
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
MeasureTheory.Measure.map id μ = μ
@[simp]
theorem
MeasureTheory.Measure.map_id'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
MeasureTheory.Measure.map (fun x => x) μ = μ
theorem
MeasureTheory.Measure.map_map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
[MeasurableSpace γ]
{μ : MeasureTheory.Measure α}
{g : β → γ}
{f : α → β}
(hg : Measurable g)
(hf : Measurable f)
:
MeasureTheory.Measure.map g (MeasureTheory.Measure.map f μ) = MeasureTheory.Measure.map (g ∘ f) μ
theorem
MeasureTheory.Measure.map_mono
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{f : α → β}
(h : μ ≤ ν)
(hf : Measurable f)
:
theorem
MeasureTheory.Measure.le_map_apply
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
(hf : AEMeasurable f)
(s : Set β)
:
↑↑μ (f ⁻¹' s) ≤ ↑↑(MeasureTheory.Measure.map f μ) s
Even if s is not measurable, we can bound map f μ s from below.
See also MeasurableEquiv.map_apply.
theorem
MeasureTheory.Measure.preimage_null_of_map_null
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
(hf : AEMeasurable f)
{s : Set β}
(hs : ↑↑(MeasureTheory.Measure.map f μ) s = 0)
:
Even if s is not measurable, map f μ s = 0 implies that μ (f ⁻¹' s) = 0.
theorem
MeasureTheory.Measure.tendsto_ae_map
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
(hf : AEMeasurable f)
:
def
MeasureTheory.Measure.comapₗ
{α : Type u_1}
{β : Type u_2}
[MeasurableSpace β]
[MeasurableSpace α]
(f : α → β)
:
Pullback of a Measure as a linear map. If f sends each measurable set to a measurable
set, then for each measurable set s we have comapₗ f μ s = μ (f '' s).
If the linearity is not needed, please use comap instead, which works for a larger class of
functions.
Instances For
theorem
MeasureTheory.Measure.comapₗ_apply
{α : Type u_1}
{s : Set α}
{β : Type u_8}
[MeasurableSpace α]
{mβ : MeasurableSpace β}
(f : α → β)
(hfi : Function.Injective f)
(hf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s))
(μ : MeasureTheory.Measure β)
(hs : MeasurableSet s)
:
↑↑(↑(MeasureTheory.Measure.comapₗ f) μ) s = ↑↑μ (f '' s)
def
MeasureTheory.Measure.comap
{α : Type u_1}
{β : Type u_2}
[MeasurableSpace β]
[MeasurableSpace α]
(f : α → β)
(μ : MeasureTheory.Measure β)
:
Pullback of a Measure. If f sends each measurable set to a null-measurable set,
then for each measurable set s we have comap f μ s = μ (f '' s).
Instances For
theorem
MeasureTheory.Measure.comap_apply₀
{α : Type u_1}
{β : Type u_2}
[MeasurableSpace β]
{s : Set α}
[MeasurableSpace α]
(f : α → β)
(μ : MeasureTheory.Measure β)
(hfi : Function.Injective f)
(hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s))
(hs : MeasureTheory.NullMeasurableSet s)
:
↑↑(MeasureTheory.Measure.comap f μ) s = ↑↑μ (f '' s)
theorem
MeasureTheory.Measure.le_comap_apply
{α : Type u_1}
{β : Type u_8}
[MeasurableSpace α]
{mβ : MeasurableSpace β}
(f : α → β)
(μ : MeasureTheory.Measure β)
(hfi : Function.Injective f)
(hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s))
(s : Set α)
:
↑↑μ (f '' s) ≤ ↑↑(MeasureTheory.Measure.comap f μ) s
theorem
MeasureTheory.Measure.comap_apply
{α : Type u_1}
{s : Set α}
{β : Type u_8}
[MeasurableSpace α]
{_mβ : MeasurableSpace β}
(f : α → β)
(hfi : Function.Injective f)
(hf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s))
(μ : MeasureTheory.Measure β)
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.comap f μ) s = ↑↑μ (f '' s)
theorem
MeasureTheory.Measure.comapₗ_eq_comap
{α : Type u_1}
{s : Set α}
{β : Type u_8}
[MeasurableSpace α]
{_mβ : MeasurableSpace β}
(f : α → β)
(hfi : Function.Injective f)
(hf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s))
(μ : MeasureTheory.Measure β)
(hs : MeasurableSet s)
:
↑↑(↑(MeasureTheory.Measure.comapₗ f) μ) s = ↑↑(MeasureTheory.Measure.comap f μ) s
theorem
MeasureTheory.Measure.measure_image_eq_zero_of_comap_eq_zero
{α : Type u_1}
{β : Type u_8}
[MeasurableSpace α]
{_mβ : MeasurableSpace β}
(f : α → β)
(μ : MeasureTheory.Measure β)
(hfi : Function.Injective f)
(hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s))
{s : Set α}
(hs : ↑↑(MeasureTheory.Measure.comap f μ) s = 0)
:
theorem
MeasureTheory.Measure.ae_eq_image_of_ae_eq_comap
{α : Type u_1}
{β : Type u_8}
[MeasurableSpace α]
{mβ : MeasurableSpace β}
(f : α → β)
(μ : MeasureTheory.Measure β)
(hfi : Function.Injective f)
(hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s))
{s : Set α}
{t : Set α}
(hst : s =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.comap f μ)] t)
:
f '' s =ᶠ[MeasureTheory.Measure.ae μ] f '' t
theorem
MeasureTheory.Measure.NullMeasurableSet.image
{α : Type u_1}
{β : Type u_8}
[MeasurableSpace α]
{mβ : MeasurableSpace β}
(f : α → β)
(μ : MeasureTheory.Measure β)
(hfi : Function.Injective f)
(hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s))
{s : Set α}
(hs : MeasureTheory.NullMeasurableSet s)
:
theorem
MeasureTheory.Measure.comap_preimage
{α : Type u_1}
{β : Type u_8}
[MeasurableSpace α]
{mβ : MeasurableSpace β}
(f : α → β)
(μ : MeasureTheory.Measure β)
{s : Set β}
(hf : Function.Injective f)
(hf' : Measurable f)
(h : ∀ (t : Set α), MeasurableSet t → MeasureTheory.NullMeasurableSet (f '' t))
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.comap f μ) (f ⁻¹' s) = ↑↑μ (s ∩ Set.range f)
Subtype of a measure space #
theorem
MeasureTheory.Measure.MeasurableSet.nullMeasurableSet_subtype_coe
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set ↑s}
(hs : MeasureTheory.NullMeasurableSet s)
(ht : MeasurableSet t)
:
MeasureTheory.NullMeasurableSet (Subtype.val '' t)
theorem
MeasureTheory.Measure.NullMeasurableSet.subtype_coe
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set ↑s}
(hs : MeasureTheory.NullMeasurableSet s)
(ht : MeasureTheory.NullMeasurableSet t)
:
MeasureTheory.NullMeasurableSet (Subtype.val '' t)
theorem
MeasureTheory.Measure.measure_subtype_coe_le_comap
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(hs : MeasureTheory.NullMeasurableSet s)
(t : Set ↑s)
:
↑↑μ (Subtype.val '' t) ≤ ↑↑(MeasureTheory.Measure.comap Subtype.val μ) t
theorem
MeasureTheory.Measure.measure_subtype_coe_eq_zero_of_comap_eq_zero
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(hs : MeasureTheory.NullMeasurableSet s)
{t : Set ↑s}
(ht : ↑↑(MeasureTheory.Measure.comap Subtype.val μ) t = 0)
:
instance
MeasureTheory.Measure.Subtype.measureSpace
{α : Type u_1}
[MeasureTheory.MeasureSpace α]
{p : α → Prop}
:
theorem
MeasureTheory.Measure.Subtype.volume_def
{α : Type u_1}
{s : Set α}
[MeasureTheory.MeasureSpace α]
:
MeasureTheory.volume = MeasureTheory.Measure.comap Subtype.val MeasureTheory.volume
theorem
MeasureTheory.Measure.Subtype.volume_univ
{α : Type u_1}
{s : Set α}
[MeasureTheory.MeasureSpace α]
(hs : MeasureTheory.NullMeasurableSet s)
:
↑↑MeasureTheory.volume Set.univ = ↑↑MeasureTheory.volume s
theorem
MeasureTheory.Measure.volume_subtype_coe_le_volume
{α : Type u_1}
{s : Set α}
[MeasureTheory.MeasureSpace α]
(hs : MeasureTheory.NullMeasurableSet s)
(t : Set ↑s)
:
theorem
MeasureTheory.Measure.volume_subtype_coe_eq_zero_of_volume_eq_zero
{α : Type u_1}
{s : Set α}
[MeasureTheory.MeasureSpace α]
(hs : MeasureTheory.NullMeasurableSet s)
{t : Set ↑s}
(ht : ↑↑MeasureTheory.volume t = 0)
:
Restricting a measure #
Restrict a measure μ to a set s as an ℝ≥0∞-linear map.
Instances For
def
MeasureTheory.Measure.restrict
{α : Type u_1}
{_m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
(s : Set α)
:
Restrict a measure μ to a set s.
Instances For
@[simp]
theorem
MeasureTheory.Measure.restrictₗ_apply
{α : Type u_1}
{_m0 : MeasurableSpace α}
(s : Set α)
(μ : MeasureTheory.Measure α)
:
theorem
MeasureTheory.Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(h : MeasurableSet s)
:
↑(MeasureTheory.Measure.restrict μ s) = ↑(MeasureTheory.OuterMeasure.restrict s) ↑μ
This lemma shows that restrict and toOuterMeasure commute. Note that the LHS has a
restrict on measures and the RHS has a restrict on outer measures.
theorem
MeasureTheory.Measure.restrict_apply₀
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(ht : MeasureTheory.NullMeasurableSet t)
:
↑↑(MeasureTheory.Measure.restrict μ s) t = ↑↑μ (t ∩ s)
@[simp]
theorem
MeasureTheory.Measure.restrict_apply
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(ht : MeasurableSet t)
:
↑↑(MeasureTheory.Measure.restrict μ s) t = ↑↑μ (t ∩ s)
If t is a measurable set, then the measure of t with respect to the restriction of
the measure to s equals the outer measure of t ∩ s. An alternate version requiring that s
be measurable instead of t exists as Measure.restrict_apply'.
theorem
MeasureTheory.Measure.restrict_mono'
{α : Type u_1}
{_m0 : MeasurableSpace α}
⦃s : Set α⦄
⦃s' : Set α⦄
⦃μ : MeasureTheory.Measure α⦄
⦃ν : MeasureTheory.Measure α⦄
(hs : s ≤ᶠ[MeasureTheory.Measure.ae μ] s')
(hμν : μ ≤ ν)
:
Restriction of a measure to a subset is monotone both in set and in measure.
theorem
MeasureTheory.Measure.restrict_mono
{α : Type u_1}
{_m0 : MeasurableSpace α}
⦃s : Set α⦄
⦃s' : Set α⦄
(hs : s ⊆ s')
⦃μ : MeasureTheory.Measure α⦄
⦃ν : MeasureTheory.Measure α⦄
(hμν : μ ≤ ν)
:
Restriction of a measure to a subset is monotone both in set and in measure.
theorem
MeasureTheory.Measure.restrict_mono_ae
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : s ≤ᶠ[MeasureTheory.Measure.ae μ] t)
:
theorem
MeasureTheory.Measure.restrict_congr_set
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : s =ᶠ[MeasureTheory.Measure.ae μ] t)
:
@[simp]
theorem
MeasureTheory.Measure.restrict_apply'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.restrict μ s) t = ↑↑μ (t ∩ s)
If s is a measurable set, then the outer measure of t with respect to the restriction of
the measure to s equals the outer measure of t ∩ s. This is an alternate version of
Measure.restrict_apply, requiring that s is measurable instead of t.
theorem
MeasureTheory.Measure.restrict_apply₀'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasureTheory.NullMeasurableSet s)
:
↑↑(MeasureTheory.Measure.restrict μ s) t = ↑↑μ (t ∩ s)
theorem
MeasureTheory.Measure.restrict_le_self
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
:
theorem
MeasureTheory.Measure.restrict_eq_self
{α : Type u_1}
{m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
{s : Set α}
{t : Set α}
(h : s ⊆ t)
:
↑↑(MeasureTheory.Measure.restrict μ t) s = ↑↑μ s
@[simp]
theorem
MeasureTheory.Measure.restrict_apply_self
{α : Type u_1}
{m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
(s : Set α)
:
↑↑(MeasureTheory.Measure.restrict μ s) s = ↑↑μ s
theorem
MeasureTheory.Measure.restrict_apply_univ
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
(s : Set α)
:
↑↑(MeasureTheory.Measure.restrict μ s) Set.univ = ↑↑μ s
theorem
MeasureTheory.Measure.le_restrict_apply
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
(s : Set α)
(t : Set α)
:
↑↑μ (t ∩ s) ≤ ↑↑(MeasureTheory.Measure.restrict μ s) t
theorem
MeasureTheory.Measure.restrict_apply_superset
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : s ⊆ t)
:
↑↑(MeasureTheory.Measure.restrict μ s) t = ↑↑μ s
@[simp]
theorem
MeasureTheory.Measure.restrict_add
{α : Type u_1}
{_m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
(ν : MeasureTheory.Measure α)
(s : Set α)
:
@[simp]
@[simp]
theorem
MeasureTheory.Measure.restrict_smul
{α : Type u_1}
{_m0 : MeasurableSpace α}
(c : ENNReal)
(μ : MeasureTheory.Measure α)
(s : Set α)
:
MeasureTheory.Measure.restrict (c • μ) s = c • MeasureTheory.Measure.restrict μ s
theorem
MeasureTheory.Measure.restrict_restrict₀
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasureTheory.NullMeasurableSet s)
:
@[simp]
theorem
MeasureTheory.Measure.restrict_restrict
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
:
theorem
MeasureTheory.Measure.restrict_restrict_of_subset
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : s ⊆ t)
:
theorem
MeasureTheory.Measure.restrict_restrict₀'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(ht : MeasureTheory.NullMeasurableSet t)
:
theorem
MeasureTheory.Measure.restrict_restrict'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(ht : MeasurableSet t)
:
theorem
MeasureTheory.Measure.restrict_comm
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
:
theorem
MeasureTheory.Measure.restrict_apply_eq_zero
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(ht : MeasurableSet t)
:
↑↑(MeasureTheory.Measure.restrict μ s) t = 0 ↔ ↑↑μ (t ∩ s) = 0
theorem
MeasureTheory.Measure.measure_inter_eq_zero_of_restrict
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : ↑↑(MeasureTheory.Measure.restrict μ s) t = 0)
:
theorem
MeasureTheory.Measure.restrict_apply_eq_zero'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.restrict μ s) t = 0 ↔ ↑↑μ (t ∩ s) = 0
@[simp]
theorem
MeasureTheory.Measure.restrict_eq_zero
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
:
MeasureTheory.Measure.restrict μ s = 0 ↔ ↑↑μ s = 0
instance
MeasureTheory.Measure.restrict.neZero
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
[NeZero (↑↑μ s)]
:
If μ s ≠ 0, then μ.restrict s ≠ 0, in terms of NeZero instances.
theorem
MeasureTheory.Measure.restrict_zero_set
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(h : ↑↑μ s = 0)
:
@[simp]
theorem
MeasureTheory.Measure.restrict_empty
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
@[simp]
theorem
MeasureTheory.Measure.restrict_univ
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
MeasureTheory.Measure.restrict μ Set.univ = μ
theorem
MeasureTheory.Measure.restrict_inter_add_diff₀
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{t : Set α}
(s : Set α)
(ht : MeasureTheory.NullMeasurableSet t)
:
MeasureTheory.Measure.restrict μ (s ∩ t) + MeasureTheory.Measure.restrict μ (s \ t) = MeasureTheory.Measure.restrict μ s
theorem
MeasureTheory.Measure.restrict_inter_add_diff
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{t : Set α}
(s : Set α)
(ht : MeasurableSet t)
:
MeasureTheory.Measure.restrict μ (s ∩ t) + MeasureTheory.Measure.restrict μ (s \ t) = MeasureTheory.Measure.restrict μ s
theorem
MeasureTheory.Measure.restrict_union_add_inter₀
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{t : Set α}
(s : Set α)
(ht : MeasureTheory.NullMeasurableSet t)
:
MeasureTheory.Measure.restrict μ (s ∪ t) + MeasureTheory.Measure.restrict μ (s ∩ t) = MeasureTheory.Measure.restrict μ s + MeasureTheory.Measure.restrict μ t
theorem
MeasureTheory.Measure.restrict_union_add_inter
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{t : Set α}
(s : Set α)
(ht : MeasurableSet t)
:
MeasureTheory.Measure.restrict μ (s ∪ t) + MeasureTheory.Measure.restrict μ (s ∩ t) = MeasureTheory.Measure.restrict μ s + MeasureTheory.Measure.restrict μ t
theorem
MeasureTheory.Measure.restrict_union_add_inter'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(hs : MeasurableSet s)
(t : Set α)
:
MeasureTheory.Measure.restrict μ (s ∪ t) + MeasureTheory.Measure.restrict μ (s ∩ t) = MeasureTheory.Measure.restrict μ s + MeasureTheory.Measure.restrict μ t
theorem
MeasureTheory.Measure.restrict_union₀
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : MeasureTheory.AEDisjoint μ s t)
(ht : MeasureTheory.NullMeasurableSet t)
:
theorem
MeasureTheory.Measure.restrict_union
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : Disjoint s t)
(ht : MeasurableSet t)
:
theorem
MeasureTheory.Measure.restrict_union'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(h : Disjoint s t)
(hs : MeasurableSet s)
:
@[simp]
theorem
MeasureTheory.Measure.restrict_add_restrict_compl
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(hs : MeasurableSet s)
:
@[simp]
theorem
MeasureTheory.Measure.restrict_compl_add_restrict
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(hs : MeasurableSet s)
:
theorem
MeasureTheory.Measure.restrict_union_le
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
(s : Set α)
(s' : Set α)
:
theorem
MeasureTheory.Measure.restrict_iUnion_apply_ae
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
(hd : Pairwise (MeasureTheory.AEDisjoint μ on s))
(hm : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i))
{t : Set α}
(ht : MeasurableSet t)
:
↑↑(MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i)) t = ∑' (i : ι), ↑↑(MeasureTheory.Measure.restrict μ (s i)) t
theorem
MeasureTheory.Measure.restrict_iUnion_apply
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
(hd : Pairwise (Disjoint on s))
(hm : ∀ (i : ι), MeasurableSet (s i))
{t : Set α}
(ht : MeasurableSet t)
:
↑↑(MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i)) t = ∑' (i : ι), ↑↑(MeasureTheory.Measure.restrict μ (s i)) t
theorem
MeasureTheory.Measure.restrict_iUnion_apply_eq_iSup
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
(hd : Directed (fun x x_1 => x ⊆ x_1) s)
{t : Set α}
(ht : MeasurableSet t)
:
↑↑(MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i)) t = ⨆ (i : ι), ↑↑(MeasureTheory.Measure.restrict μ (s i)) t
theorem
MeasureTheory.Measure.restrict_map
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
(hf : Measurable f)
{s : Set β}
(hs : MeasurableSet s)
:
The restriction of the pushforward measure is the pushforward of the restriction. For a version
assuming only AEMeasurable, see restrict_map_of_aemeasurable.
theorem
MeasureTheory.Measure.restrict_toMeasurable
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(h : ↑↑μ s ≠ ⊤)
:
theorem
MeasureTheory.Measure.restrict_eq_self_of_ae_mem
{α : Type u_1}
{_m0 : MeasurableSpace α}
⦃s : Set α⦄
⦃μ : MeasureTheory.Measure α⦄
(hs : ∀ᵐ (x : α) ∂μ, x ∈ s)
:
theorem
MeasureTheory.Measure.restrict_congr_meas
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set α}
(hs : MeasurableSet s)
:
MeasureTheory.Measure.restrict μ s = MeasureTheory.Measure.restrict ν s ↔ ∀ (t : Set α), t ⊆ s → MeasurableSet t → ↑↑μ t = ↑↑ν t
theorem
MeasureTheory.Measure.restrict_congr_mono
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
(hs : s ⊆ t)
(h : MeasureTheory.Measure.restrict μ t = MeasureTheory.Measure.restrict ν t)
:
theorem
MeasureTheory.Measure.restrict_union_congr
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set α}
{t : Set α}
:
If two measures agree on all measurable subsets of s and t, then they agree on all
measurable subsets of s ∪ t.
theorem
MeasureTheory.Measure.restrict_finset_biUnion_congr
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Finset ι}
{t : ι → Set α}
:
MeasureTheory.Measure.restrict μ (⋃ (i : ι) (_ : i ∈ s), t i) = MeasureTheory.Measure.restrict ν (⋃ (i : ι) (_ : i ∈ s), t i) ↔ ∀ (i : ι), i ∈ s → MeasureTheory.Measure.restrict μ (t i) = MeasureTheory.Measure.restrict ν (t i)
theorem
MeasureTheory.Measure.restrict_iUnion_congr
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
:
MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i) = MeasureTheory.Measure.restrict ν (⋃ (i : ι), s i) ↔ ∀ (i : ι), MeasureTheory.Measure.restrict μ (s i) = MeasureTheory.Measure.restrict ν (s i)
theorem
MeasureTheory.Measure.restrict_biUnion_congr
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set ι}
{t : ι → Set α}
(hc : Set.Countable s)
:
MeasureTheory.Measure.restrict μ (⋃ (i : ι) (_ : i ∈ s), t i) = MeasureTheory.Measure.restrict ν (⋃ (i : ι) (_ : i ∈ s), t i) ↔ ∀ (i : ι), i ∈ s → MeasureTheory.Measure.restrict μ (t i) = MeasureTheory.Measure.restrict ν (t i)
theorem
MeasureTheory.Measure.restrict_sUnion_congr
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{S : Set (Set α)}
(hc : Set.Countable S)
:
MeasureTheory.Measure.restrict μ (⋃₀ S) = MeasureTheory.Measure.restrict ν (⋃₀ S) ↔ ∀ (s : Set α), s ∈ S → MeasureTheory.Measure.restrict μ s = MeasureTheory.Measure.restrict ν s
theorem
MeasureTheory.Measure.restrict_sInf_eq_sInf_restrict
{α : Type u_1}
{t : Set α}
{m0 : MeasurableSpace α}
{m : Set (MeasureTheory.Measure α)}
(hm : Set.Nonempty m)
(ht : MeasurableSet t)
:
MeasureTheory.Measure.restrict (sInf m) t = sInf ((fun μ => MeasureTheory.Measure.restrict μ t) '' m)
This lemma shows that Inf and restrict commute for measures.
theorem
MeasureTheory.Measure.exists_mem_of_measure_ne_zero_of_ae
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
(hs : ↑↑μ s ≠ 0)
{p : α → Prop}
(hp : ∀ᵐ (x : α) ∂MeasureTheory.Measure.restrict μ s, p x)
:
Extensionality results #
theorem
MeasureTheory.Measure.ext_iff_of_iUnion_eq_univ
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
(hs : ⋃ (i : ι), s i = Set.univ)
:
μ = ν ↔ ∀ (i : ι), MeasureTheory.Measure.restrict μ (s i) = MeasureTheory.Measure.restrict ν (s i)
Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using Union).
theorem
MeasureTheory.Measure.ext_of_iUnion_eq_univ
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
(hs : ⋃ (i : ι), s i = Set.univ)
:
(∀ (i : ι), MeasureTheory.Measure.restrict μ (s i) = MeasureTheory.Measure.restrict ν (s i)) → μ = ν
Alias of the reverse direction of MeasureTheory.Measure.ext_iff_of_iUnion_eq_univ.
Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using Union).
theorem
MeasureTheory.Measure.ext_iff_of_biUnion_eq_univ
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{S : Set ι}
{s : ι → Set α}
(hc : Set.Countable S)
(hs : ⋃ (i : ι) (_ : i ∈ S), s i = Set.univ)
:
μ = ν ↔ ∀ (i : ι), i ∈ S → MeasureTheory.Measure.restrict μ (s i) = MeasureTheory.Measure.restrict ν (s i)
Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using biUnion).
theorem
MeasureTheory.Measure.ext_of_biUnion_eq_univ
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{S : Set ι}
{s : ι → Set α}
(hc : Set.Countable S)
(hs : ⋃ (i : ι) (_ : i ∈ S), s i = Set.univ)
:
(∀ (i : ι), i ∈ S → MeasureTheory.Measure.restrict μ (s i) = MeasureTheory.Measure.restrict ν (s i)) → μ = ν
Alias of the reverse direction of MeasureTheory.Measure.ext_iff_of_biUnion_eq_univ.
Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using biUnion).
theorem
MeasureTheory.Measure.ext_iff_of_sUnion_eq_univ
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{S : Set (Set α)}
(hc : Set.Countable S)
(hs : ⋃₀ S = Set.univ)
:
μ = ν ↔ ∀ (s : Set α), s ∈ S → MeasureTheory.Measure.restrict μ s = MeasureTheory.Measure.restrict ν s
Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using sUnion).
theorem
MeasureTheory.Measure.ext_of_sUnion_eq_univ
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{S : Set (Set α)}
(hc : Set.Countable S)
(hs : ⋃₀ S = Set.univ)
:
(∀ (s : Set α), s ∈ S → MeasureTheory.Measure.restrict μ s = MeasureTheory.Measure.restrict ν s) → μ = ν
Alias of the reverse direction of MeasureTheory.Measure.ext_iff_of_sUnion_eq_univ.
Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using sUnion).
theorem
MeasureTheory.Measure.ext_of_generateFrom_of_cover
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{S : Set (Set α)}
{T : Set (Set α)}
(h_gen : m0 = MeasurableSpace.generateFrom S)
(hc : Set.Countable T)
(h_inter : IsPiSystem S)
(hU : ⋃₀ T = Set.univ)
(htop : ∀ (t : Set α), t ∈ T → ↑↑μ t ≠ ⊤)
(ST_eq : ∀ (t : Set α), t ∈ T → ∀ (s : Set α), s ∈ S → ↑↑μ (s ∩ t) = ↑↑ν (s ∩ t))
(T_eq : ∀ (t : Set α), t ∈ T → ↑↑μ t = ↑↑ν t)
:
μ = ν
theorem
MeasureTheory.Measure.ext_of_generateFrom_of_cover_subset
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{S : Set (Set α)}
{T : Set (Set α)}
(h_gen : m0 = MeasurableSpace.generateFrom S)
(h_inter : IsPiSystem S)
(h_sub : T ⊆ S)
(hc : Set.Countable T)
(hU : ⋃₀ T = Set.univ)
(htop : ∀ (s : Set α), s ∈ T → ↑↑μ s ≠ ⊤)
(h_eq : ∀ (s : Set α), s ∈ S → ↑↑μ s = ↑↑ν s)
:
μ = ν
Two measures are equal if they are equal on the π-system generating the σ-algebra,
and they are both finite on an increasing spanning sequence of sets in the π-system.
This lemma is formulated using sUnion.
theorem
MeasureTheory.Measure.ext_of_generateFrom_of_iUnion
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
(C : Set (Set α))
(B : ℕ → Set α)
(hA : m0 = MeasurableSpace.generateFrom C)
(hC : IsPiSystem C)
(h1B : ⋃ (i : ℕ), B i = Set.univ)
(h2B : ∀ (i : ℕ), B i ∈ C)
(hμB : ∀ (i : ℕ), ↑↑μ (B i) ≠ ⊤)
(h_eq : ∀ (s : Set α), s ∈ C → ↑↑μ s = ↑↑ν s)
:
μ = ν
Two measures are equal if they are equal on the π-system generating the σ-algebra,
and they are both finite on an increasing spanning sequence of sets in the π-system.
This lemma is formulated using iUnion.
FiniteSpanningSetsIn.ext is a reformulation of this lemma.
def
MeasureTheory.Measure.sum
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
(f : ι → MeasureTheory.Measure α)
:
Sum of an indexed family of measures.
Instances For
theorem
MeasureTheory.Measure.le_sum_apply
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
(f : ι → MeasureTheory.Measure α)
(s : Set α)
:
∑' (i : ι), ↑↑(f i) s ≤ ↑↑(MeasureTheory.Measure.sum f) s
@[simp]
theorem
MeasureTheory.Measure.sum_apply
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
(f : ι → MeasureTheory.Measure α)
{s : Set α}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.sum f) s = ∑' (i : ι), ↑↑(f i) s
theorem
MeasureTheory.Measure.le_sum
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
(μ : ι → MeasureTheory.Measure α)
(i : ι)
:
μ i ≤ MeasureTheory.Measure.sum μ
@[simp]
theorem
MeasureTheory.Measure.sum_apply_eq_zero
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
[Countable ι]
{μ : ι → MeasureTheory.Measure α}
{s : Set α}
:
↑↑(MeasureTheory.Measure.sum μ) s = 0 ↔ ∀ (i : ι), ↑↑(μ i) s = 0
theorem
MeasureTheory.Measure.sum_apply_eq_zero'
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : ι → MeasureTheory.Measure α}
{s : Set α}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.sum μ) s = 0 ↔ ∀ (i : ι), ↑↑(μ i) s = 0
theorem
MeasureTheory.Measure.sum_comm
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{ι' : Type u_8}
(μ : ι → ι' → MeasureTheory.Measure α)
:
(MeasureTheory.Measure.sum fun n => MeasureTheory.Measure.sum (μ n)) = MeasureTheory.Measure.sum fun m => MeasureTheory.Measure.sum fun n => μ n m
theorem
MeasureTheory.Measure.ae_sum_iff
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
[Countable ι]
{μ : ι → MeasureTheory.Measure α}
{p : α → Prop}
:
(∀ᵐ (x : α) ∂MeasureTheory.Measure.sum μ, p x) ↔ ∀ (i : ι), ∀ᵐ (x : α) ∂μ i, p x
theorem
MeasureTheory.Measure.ae_sum_iff'
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : ι → MeasureTheory.Measure α}
{p : α → Prop}
(h : MeasurableSet {x | p x})
:
(∀ᵐ (x : α) ∂MeasureTheory.Measure.sum μ, p x) ↔ ∀ (i : ι), ∀ᵐ (x : α) ∂μ i, p x
@[simp]
theorem
MeasureTheory.Measure.sum_fintype
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
[Fintype ι]
(μ : ι → MeasureTheory.Measure α)
:
MeasureTheory.Measure.sum μ = Finset.sum Finset.univ fun i => μ i
@[simp]
theorem
MeasureTheory.Measure.sum_coe_finset
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
(s : Finset ι)
(μ : ι → MeasureTheory.Measure α)
:
(MeasureTheory.Measure.sum fun i => μ ↑i) = Finset.sum s fun i => μ i
@[simp]
theorem
MeasureTheory.Measure.ae_sum_eq
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
[Countable ι]
(μ : ι → MeasureTheory.Measure α)
:
MeasureTheory.Measure.ae (MeasureTheory.Measure.sum μ) = ⨆ (i : ι), MeasureTheory.Measure.ae (μ i)
theorem
MeasureTheory.Measure.sum_bool
{α : Type u_1}
{m0 : MeasurableSpace α}
(f : Bool → MeasureTheory.Measure α)
:
MeasureTheory.Measure.sum f = f true + f false
theorem
MeasureTheory.Measure.sum_cond
{α : Type u_1}
{m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
(ν : MeasureTheory.Measure α)
:
(MeasureTheory.Measure.sum fun b => bif b then μ else ν) = μ + ν
@[simp]
theorem
MeasureTheory.Measure.restrict_sum
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
(μ : ι → MeasureTheory.Measure α)
{s : Set α}
(hs : MeasurableSet s)
:
theorem
MeasureTheory.Measure.sum_of_empty
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
[IsEmpty ι]
(μ : ι → MeasureTheory.Measure α)
:
theorem
MeasureTheory.Measure.sum_add_sum_compl
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
(s : Set ι)
(μ : ι → MeasureTheory.Measure α)
:
((MeasureTheory.Measure.sum fun i => μ ↑i) + MeasureTheory.Measure.sum fun i => μ ↑i) = MeasureTheory.Measure.sum μ
theorem
MeasureTheory.Measure.sum_congr
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : ℕ → MeasureTheory.Measure α}
{ν : ℕ → MeasureTheory.Measure α}
(h : ∀ (n : ℕ), μ n = ν n)
:
theorem
MeasureTheory.Measure.sum_add_sum
{α : Type u_1}
{m0 : MeasurableSpace α}
(μ : ℕ → MeasureTheory.Measure α)
(ν : ℕ → MeasureTheory.Measure α)
:
MeasureTheory.Measure.sum μ + MeasureTheory.Measure.sum ν = MeasureTheory.Measure.sum fun n => μ n + ν n
theorem
MeasureTheory.Measure.restrict_iUnion_ae
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
(hd : Pairwise (MeasureTheory.AEDisjoint μ on s))
(hm : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i))
:
MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i) = MeasureTheory.Measure.sum fun i => MeasureTheory.Measure.restrict μ (s i)
theorem
MeasureTheory.Measure.restrict_iUnion
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
(hd : Pairwise (Disjoint on s))
(hm : ∀ (i : ι), MeasurableSet (s i))
:
MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i) = MeasureTheory.Measure.sum fun i => MeasureTheory.Measure.restrict μ (s i)
theorem
MeasureTheory.Measure.restrict_iUnion_le
{α : Type u_1}
{ι : Type u_5}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[Countable ι]
{s : ι → Set α}
:
MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i) ≤ MeasureTheory.Measure.sum fun i => MeasureTheory.Measure.restrict μ (s i)
Absolute continuity #
def
MeasureTheory.Measure.AbsolutelyContinuous
{α : Type u_1}
{_m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
(ν : MeasureTheory.Measure α)
:
We say that μ is absolutely continuous with respect to ν, or that μ is dominated by ν,
if ν(A) = 0 implies that μ(A) = 0.
Instances For
We say that μ is absolutely continuous with respect to ν, or that μ is dominated by ν,
if ν(A) = 0 implies that μ(A) = 0.
Instances For
theorem
MeasureTheory.Measure.absolutelyContinuous_of_le
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
(h : μ ≤ ν)
:
theorem
LE.le.absolutelyContinuous
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
(h : μ ≤ ν)
:
theorem
MeasureTheory.Measure.absolutelyContinuous_of_eq
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
(h : μ = ν)
:
theorem
Eq.absolutelyContinuous
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
(h : μ = ν)
:
theorem
MeasureTheory.Measure.AbsolutelyContinuous.mk
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
(h : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑ν s = 0 → ↑↑μ s = 0)
:
theorem
MeasureTheory.Measure.AbsolutelyContinuous.refl
{α : Type u_1}
{_m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
:
theorem
MeasureTheory.Measure.AbsolutelyContinuous.rfl
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
instance
MeasureTheory.Measure.AbsolutelyContinuous.instIsRefl
{α : Type u_1}
[MeasurableSpace α]
:
IsRefl (MeasureTheory.Measure α) fun x x_1 => MeasureTheory.Measure.AbsolutelyContinuous x x_1
theorem
MeasureTheory.Measure.AbsolutelyContinuous.trans
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ₁ : MeasureTheory.Measure α}
{μ₂ : MeasureTheory.Measure α}
{μ₃ : MeasureTheory.Measure α}
(h1 : MeasureTheory.Measure.AbsolutelyContinuous μ₁ μ₂)
(h2 : MeasureTheory.Measure.AbsolutelyContinuous μ₂ μ₃)
:
theorem
MeasureTheory.Measure.AbsolutelyContinuous.map
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
(h : MeasureTheory.Measure.AbsolutelyContinuous μ ν)
{f : α → β}
(hf : Measurable f)
:
theorem
MeasureTheory.Measure.AbsolutelyContinuous.smul
{α : Type u_1}
{R : Type u_6}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[Monoid R]
[DistribMulAction R ENNReal]
[IsScalarTower R ENNReal ENNReal]
(h : MeasureTheory.Measure.AbsolutelyContinuous μ ν)
(c : R)
:
theorem
MeasureTheory.Measure.absolutelyContinuous_of_le_smul
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{μ' : MeasureTheory.Measure α}
{c : ENNReal}
(hμ'_le : μ' ≤ c • μ)
:
theorem
MeasureTheory.Measure.ae_le_iff_absolutelyContinuous
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
:
theorem
LE.le.absolutelyContinuous_of_ae
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
:
Alias of the forward direction of MeasureTheory.Measure.ae_le_iff_absolutelyContinuous.
theorem
MeasureTheory.Measure.AbsolutelyContinuous.ae_le
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
:
Alias of the reverse direction of MeasureTheory.Measure.ae_le_iff_absolutelyContinuous.
theorem
MeasureTheory.Measure.ae_mono'
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
:
Alias of the reverse direction of MeasureTheory.Measure.ae_le_iff_absolutelyContinuous.
Alias of the reverse direction of MeasureTheory.Measure.ae_le_iff_absolutelyContinuous.
theorem
MeasureTheory.Measure.AbsolutelyContinuous.ae_eq
{α : Type u_1}
{δ : Type u_4}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
(h : MeasureTheory.Measure.AbsolutelyContinuous μ ν)
{f : α → δ}
{g : α → δ}
(h' : f =ᶠ[MeasureTheory.Measure.ae ν] g)
:
Quasi measure preserving maps (a.k.a. non-singular maps) #
structure
MeasureTheory.Measure.QuasiMeasurePreserving
{α : Type u_1}
{β : Type u_2}
[MeasurableSpace β]
{m0 : MeasurableSpace α}
(f : α → β)
(μa : autoParam (MeasureTheory.Measure α) _auto✝)
(μb : autoParam (MeasureTheory.Measure β) _auto✝)
:
- measurable : Measurable f
- absolutelyContinuous : MeasureTheory.Measure.AbsolutelyContinuous (MeasureTheory.Measure.map f μa) μb
A map f : α → β is said to be quasi measure preserving (a.k.a. non-singular) w.r.t. measures
μa and μb if it is measurable and μb s = 0 implies μa (f ⁻¹' s) = 0.
Instances For
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.id
{α : Type u_1}
{_m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
:
theorem
Measurable.quasiMeasurePreserving
{α : Type u_1}
{β : Type u_2}
[MeasurableSpace β]
{f : α → β}
{_m0 : MeasurableSpace α}
(hf : Measurable f)
(μ : MeasureTheory.Measure α)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.mono_left
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μa' : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{f : α → β}
(h : MeasureTheory.Measure.QuasiMeasurePreserving f)
(ha : MeasureTheory.Measure.AbsolutelyContinuous μa' μa)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.mono_right
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{μb' : MeasureTheory.Measure β}
{f : α → β}
(h : MeasureTheory.Measure.QuasiMeasurePreserving f)
(ha : MeasureTheory.Measure.AbsolutelyContinuous μb μb')
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.mono
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μa' : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{μb' : MeasureTheory.Measure β}
{f : α → β}
(ha : MeasureTheory.Measure.AbsolutelyContinuous μa' μa)
(hb : MeasureTheory.Measure.AbsolutelyContinuous μb μb')
(h : MeasureTheory.Measure.QuasiMeasurePreserving f)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.comp
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
[MeasurableSpace γ]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{μc : MeasureTheory.Measure γ}
{g : β → γ}
{f : α → β}
(hg : MeasureTheory.Measure.QuasiMeasurePreserving g)
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.iterate
{α : Type u_1}
{m0 : MeasurableSpace α}
{μa : MeasureTheory.Measure α}
{f : α → α}
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
(n : ℕ)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.aemeasurable
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{f : α → β}
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.ae_map_le
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{f : α → β}
(h : MeasureTheory.Measure.QuasiMeasurePreserving f)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.tendsto_ae
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{f : α → β}
(h : MeasureTheory.Measure.QuasiMeasurePreserving f)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.ae
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{f : α → β}
(h : MeasureTheory.Measure.QuasiMeasurePreserving f)
{p : β → Prop}
(hg : ∀ᵐ (x : β) ∂μb, p x)
:
∀ᵐ (x : α) ∂μa, p (f x)
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.ae_eq
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{f : α → β}
(h : MeasureTheory.Measure.QuasiMeasurePreserving f)
{g₁ : β → δ}
{g₂ : β → δ}
(hg : g₁ =ᶠ[MeasureTheory.Measure.ae μb] g₂)
:
g₁ ∘ f =ᶠ[MeasureTheory.Measure.ae μa] g₂ ∘ f
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.preimage_null
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{f : α → β}
(h : MeasureTheory.Measure.QuasiMeasurePreserving f)
{s : Set β}
(hs : ↑↑μb s = 0)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.preimage_mono_ae
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{f : α → β}
{s : Set β}
{t : Set β}
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
(h : s ≤ᶠ[MeasureTheory.Measure.ae μb] t)
:
f ⁻¹' s ≤ᶠ[MeasureTheory.Measure.ae μa] f ⁻¹' t
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.preimage_ae_eq
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{f : α → β}
{s : Set β}
{t : Set β}
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
(h : s =ᶠ[MeasureTheory.Measure.ae μb] t)
:
f ⁻¹' s =ᶠ[MeasureTheory.Measure.ae μa] f ⁻¹' t
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.preimage_iterate_ae_eq
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{f : α → α}
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
(k : ℕ)
(hs : f ⁻¹' s =ᶠ[MeasureTheory.Measure.ae μ] s)
:
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.image_zpow_ae_eq
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{e : α ≃ α}
(he : MeasureTheory.Measure.QuasiMeasurePreserving ↑e)
(he' : MeasureTheory.Measure.QuasiMeasurePreserving ↑e.symm)
(k : ℤ)
(hs : ↑e '' s =ᶠ[MeasureTheory.Measure.ae μ] s)
:
↑(e ^ k) '' s =ᶠ[MeasureTheory.Measure.ae μ] s
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.limsup_preimage_iterate_ae_eq
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{f : α → α}
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
(hs : f ⁻¹' s =ᶠ[MeasureTheory.Measure.ae μ] s)
:
Filter.limsup (fun n => (Set.preimage f)^[n] s) Filter.atTop =ᶠ[MeasureTheory.Measure.ae μ] s
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.liminf_preimage_iterate_ae_eq
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{f : α → α}
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
(hs : f ⁻¹' s =ᶠ[MeasureTheory.Measure.ae μ] s)
:
Filter.liminf (fun n => (Set.preimage f)^[n] s) Filter.atTop =ᶠ[MeasureTheory.Measure.ae μ] s
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.exists_preimage_eq_of_preimage_ae
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
{f : α → α}
(h : MeasureTheory.Measure.QuasiMeasurePreserving f)
(hs : MeasurableSet s)
(hs' : f ⁻¹' s =ᶠ[MeasureTheory.Measure.ae μ] s)
:
∃ t, MeasurableSet t ∧ t =ᶠ[MeasureTheory.Measure.ae μ] s ∧ f ⁻¹' t = t
By replacing a measurable set that is almost invariant with the limsup of its preimages, we
obtain a measurable set that is almost equal and strictly invariant.
(The liminf would work just as well.)
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.vadd_ae_eq_of_ae_eq
{G : Type u_8}
{α : Type u_9}
[AddGroup G]
[AddAction G α]
[MeasurableSpace α]
{s : Set α}
{t : Set α}
{μ : MeasureTheory.Measure α}
(g : G)
(h_qmp : MeasureTheory.Measure.QuasiMeasurePreserving ((fun x x_1 => x +ᵥ x_1) (-g)))
(h_ae_eq : s =ᶠ[MeasureTheory.Measure.ae μ] t)
:
g +ᵥ s =ᶠ[MeasureTheory.Measure.ae μ] g +ᵥ t
theorem
MeasureTheory.Measure.QuasiMeasurePreserving.smul_ae_eq_of_ae_eq
{G : Type u_8}
{α : Type u_9}
[Group G]
[MulAction G α]
[MeasurableSpace α]
{s : Set α}
{t : Set α}
{μ : MeasureTheory.Measure α}
(g : G)
(h_qmp : MeasureTheory.Measure.QuasiMeasurePreserving ((fun x x_1 => x • x_1) g⁻¹))
(h_ae_eq : s =ᶠ[MeasureTheory.Measure.ae μ] t)
:
g • s =ᶠ[MeasureTheory.Measure.ae μ] g • t
theorem
MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_zero
{G : Type u_8}
{α : Type u_9}
[AddGroup G]
[AddAction G α]
[MeasurableSpace α]
{μ : MeasureTheory.Measure α}
{s : Set α}
(h_ae_disjoint : ∀ (g : G), g ≠ 0 → MeasureTheory.AEDisjoint μ (g +ᵥ s) s)
(h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving ((fun x x_1 => x +ᵥ x_1) g))
:
Pairwise (MeasureTheory.AEDisjoint μ on fun g => g +ᵥ s)
theorem
MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_one
{G : Type u_8}
{α : Type u_9}
[Group G]
[MulAction G α]
[MeasurableSpace α]
{μ : MeasureTheory.Measure α}
{s : Set α}
(h_ae_disjoint : ∀ (g : G), g ≠ 1 → MeasureTheory.AEDisjoint μ (g • s) s)
(h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving ((fun x x_1 => x • x_1) g))
:
Pairwise (MeasureTheory.AEDisjoint μ on fun g => g • s)
def
MeasureTheory.Measure.cofinite
{α : Type u_1}
{m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
:
Filter α
The filter of sets s such that sᶜ has finite measure.
Instances For
theorem
MeasureTheory.Measure.mem_cofinite
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
:
theorem
MeasureTheory.Measure.compl_mem_cofinite
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{s : Set α}
:
theorem
MeasureTheory.Measure.eventually_cofinite
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{p : α → Prop}
:
(∀ᶠ (x : α) in MeasureTheory.Measure.cofinite μ, p x) ↔ ↑↑μ {x | ¬p x} < ⊤
theorem
MeasureTheory.NullMeasurableSet.preimage
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
{f : α → β}
{t : Set β}
(ht : MeasureTheory.NullMeasurableSet t)
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
:
The preimage of a null measurable set under a (quasi) measure preserving map is a null measurable set.
theorem
MeasureTheory.NullMeasurableSet.mono_ac
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set α}
(h : MeasureTheory.NullMeasurableSet s)
(hle : MeasureTheory.Measure.AbsolutelyContinuous ν μ)
:
theorem
MeasureTheory.NullMeasurableSet.mono
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
{s : Set α}
(h : MeasureTheory.NullMeasurableSet s)
(hle : ν ≤ μ)
:
theorem
MeasureTheory.AEDisjoint.preimage
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
{f : α → β}
{s : Set β}
{t : Set β}
(ht : MeasureTheory.AEDisjoint ν s t)
(hf : MeasureTheory.Measure.QuasiMeasurePreserving f)
:
MeasureTheory.AEDisjoint μ (f ⁻¹' s) (f ⁻¹' t)
@[simp]
theorem
MeasureTheory.ae_eq_bot
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
MeasureTheory.Measure.ae μ = ⊥ ↔ μ = 0
@[simp]
theorem
MeasureTheory.ae_neBot
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
:
Filter.NeBot (MeasureTheory.Measure.ae μ) ↔ μ ≠ 0
instance
MeasureTheory.Measure.ae.neBot
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NeZero μ]
:
@[simp]
theorem
MeasureTheory.ae_mono
{α : Type u_1}
{m0 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
(h : μ ≤ ν)
:
theorem
MeasureTheory.mem_ae_map_iff
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{f : α → β}
(hf : AEMeasurable f)
{s : Set β}
(hs : MeasurableSet s)
:
s ∈ MeasureTheory.Measure.ae (MeasureTheory.Measure.map f μ) ↔ f ⁻¹' s ∈ MeasureTheory.Measure.ae μ
theorem
MeasureTheory.mem_ae_of_mem_ae_map
{α : Type u_1}
{β : Type u_2}
{m0 : MeasurableSpace α}
[MeasurableSpace β]
{μ :