# Documentation

Mathlib.MeasureTheory.Measure.MeasureSpace

# 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:

1. μ ∅ = 0;
2. μ is 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 cover univ where μ is 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 #

• completion is the completion of a measure to all null measurable sets.
• Measure.ofMeasurable and OuterMeasure.toMeasure are 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.ofMeasurable is a way to define a measure by only giving its value on measurable sets and proving the properties (1) and (2) mentioned above.
• OuterMeasure.toMeasure is 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 general ext_of_generateFrom_of_cover
• ext_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 of ext_of_generateFrom_of_iUnion using C ∪ {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.

## Tags #

measure, almost everywhere, measure space, completion, null set, null measurable set

instance MeasureTheory.ae_isMeasurablyGenerated {α : Type u_1} {m : } {μ : } :
theorem MeasureTheory.ae_uIoc_iff {α : Type u_1} {m : } {μ : } [] {a : α} {b : α} {P : αProp} :
(∀ᵐ (x : α) ∂μ, x Ι a bP x) (∀ᵐ (x : α) ∂μ, x Set.Ioc a bP x) ∀ᵐ (x : α) ∂μ, x Set.Ioc b aP x

See also MeasureTheory.ae_restrict_uIoc_iff.

theorem MeasureTheory.measure_union {α : Type u_1} {m : } {μ : } {s₁ : Set α} {s₂ : Set α} (hd : Disjoint s₁ s₂) (h : ) :
μ (s₁ s₂) = μ s₁ + μ s₂
theorem MeasureTheory.measure_union' {α : Type u_1} {m : } {μ : } {s₁ : Set α} {s₂ : Set α} (hd : Disjoint s₁ s₂) (h : ) :
μ (s₁ s₂) = μ s₁ + μ s₂
theorem MeasureTheory.measure_inter_add_diff {α : Type u_1} {m : } {μ : } {t : Set α} (s : Set α) (ht : ) :
μ (s t) + μ (s \ t) = μ s
theorem MeasureTheory.measure_diff_add_inter {α : Type u_1} {m : } {μ : } {t : Set α} (s : Set α) (ht : ) :
μ (s \ t) + μ (s t) = μ s
theorem MeasureTheory.measure_union_add_inter {α : Type u_1} {m : } {μ : } {t : Set α} (s : Set α) (ht : ) :
μ (s t) + μ (s t) = μ s + μ t
theorem MeasureTheory.measure_union_add_inter' {α : Type u_1} {m : } {μ : } {s : Set α} (hs : ) (t : Set α) :
μ (s t) + μ (s t) = μ s + μ t
theorem MeasureTheory.measure_symmDiff_eq {α : Type u_1} {m : } {μ : } {s : Set α} {t : Set α} (hs : ) (ht : ) :
μ (s t) = μ (s \ t) + μ (t \ s)
theorem MeasureTheory.measure_symmDiff_le {α : Type u_1} {m : } {μ : } (s : Set α) (t : Set α) (u : Set α) :
μ (s u) μ (s t) + μ (t u)
theorem MeasureTheory.measure_add_measure_compl {α : Type u_1} {m : } {μ : } {s : Set α} (h : ) :
μ s + μ s = μ Set.univ
theorem MeasureTheory.measure_biUnion₀ {α : Type u_1} {β : Type u_2} {m : } {μ : } {s : Set β} {f : βSet α} (hs : ) (hd : ) (h : ∀ (b : β), b s) :
μ (⋃ (b : β) (_ : b s), f b) = ∑' (p : s), μ (f p)
theorem MeasureTheory.measure_biUnion {α : Type u_1} {β : Type u_2} {m : } {μ : } {s : Set β} {f : βSet α} (hs : ) (hd : ) (h : ∀ (b : β), b sMeasurableSet (f b)) :
μ (⋃ (b : β) (_ : b s), f b) = ∑' (p : s), μ (f p)
theorem MeasureTheory.measure_sUnion₀ {α : Type u_1} {m : } {μ : } {S : Set (Set α)} (hs : ) (hd : ) (h : ∀ (s : Set α), ) :
μ (⋃₀ S) = ∑' (s : S), μ s
theorem MeasureTheory.measure_sUnion {α : Type u_1} {m : } {μ : } {S : Set (Set α)} (hs : ) (hd : Set.Pairwise S Disjoint) (h : ∀ (s : Set α), s S) :
μ (⋃₀ S) = ∑' (s : S), μ s
theorem MeasureTheory.measure_biUnion_finset₀ {α : Type u_1} {ι : Type u_5} {m : } {μ : } {s : } {f : ιSet α} (hd : Set.Pairwise (s) ()) (hm : ∀ (b : ι), b s) :
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 : } {μ : } {s : } {f : ιSet α} (hd : Set.PairwiseDisjoint (s) f) (hm : ∀ (b : ι), b sMeasurableSet (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} [] (μ : ) {As : ιSet α} (As_mble : ∀ (i : ι), ) (As_disj : ) :
∑' (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} [] (μ : ) {As : ιSet 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 : } {μ : } {s : Set β} (hs : ) {f : αβ} (hf : ∀ (y : β), y sMeasurableSet (f ⁻¹' {y})) :
∑' (b : s), μ (f ⁻¹' {b}) = μ (f ⁻¹' s)

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 : } {μ : } (s : ) {f : αβ} (hf : ∀ (y : β), y sMeasurableSet (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 : } {μ : } {s₁ : Set α} {s₂ : Set α} (h : μ (s₁ s₂) = 0) :
μ (s₁ \ s₂) = μ s₁
theorem MeasureTheory.measure_diff_null {α : Type u_1} {m : } {μ : } {s₁ : Set α} {s₂ : Set α} (h : μ s₂ = 0) :
μ (s₁ \ s₂) = μ s₁
theorem MeasureTheory.measure_add_diff {α : Type u_1} {m : } {μ : } {s : Set α} (hs : ) (t : Set α) :
μ s + μ (t \ s) = μ (s t)
theorem MeasureTheory.measure_diff' {α : Type u_1} {m : } {μ : } {t : Set α} (s : Set α) (hm : ) (h_fin : μ t ) :
μ (s \ t) = μ (s t) - μ t
theorem MeasureTheory.measure_diff {α : Type u_1} {m : } {μ : } {s₁ : Set α} {s₂ : Set α} (h : s₂ s₁) (h₂ : ) (h_fin : μ s₂ ) :
μ (s₁ \ s₂) = μ s₁ - μ s₂
theorem MeasureTheory.le_measure_diff {α : Type u_1} {m : } {μ : } {s₁ : Set α} {s₂ : Set α} :
μ s₁ - μ s₂ μ (s₁ \ s₂)
theorem MeasureTheory.measure_diff_lt_of_lt_add {α : Type u_1} {m : } {μ : } {s : Set α} {t : Set α} (hs : ) (hst : s t) (hs' : μ s ) {ε : ENNReal} (h : μ t < μ s + ε) :
μ (t \ s) < ε
theorem MeasureTheory.measure_diff_le_iff_le_add {α : Type u_1} {m : } {μ : } {s : Set α} {t : Set α} (hs : ) (hst : s t) (hs' : μ s ) {ε : ENNReal} :
μ (t \ s) ε μ t μ s + ε
theorem MeasureTheory.measure_eq_measure_of_null_diff {α : Type u_1} {m : } {μ : } {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 : } {μ : } {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 μ s₂ = μ s₃
theorem MeasureTheory.measure_eq_measure_smaller_of_between_null_diff {α : Type u_1} {m : } {μ : } {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 : } {μ : } {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 : } {μ : } {s : Set α} (h₁ : ) (h_fin : μ s ) :
μ s = μ Set.univ - μ s
@[simp]
theorem MeasureTheory.union_ae_eq_left_iff_ae_subset {α : Type u_1} {m : } {μ : } {s : Set α} {t : Set α} :
@[simp]
theorem MeasureTheory.union_ae_eq_right_iff_ae_subset {α : Type u_1} {m : } {μ : } {s : Set α} {t : Set α} :
theorem MeasureTheory.ae_eq_of_ae_subset_of_measure_ge {α : Type u_1} {m : } {μ : } {s : Set α} {t : Set α} (h₁ : ) (h₂ : μ t μ s) (hsm : ) (ht : μ t ) :
theorem MeasureTheory.ae_eq_of_subset_of_measure_ge {α : Type u_1} {m : } {μ : } {s : Set α} {t : Set α} (h₁ : s t) (h₂ : μ t μ s) (hsm : ) (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 : } {μ : } [] {s : βSet α} {t : βSet 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)
theorem MeasureTheory.measure_union_congr_of_subset {α : Type u_1} {m : } {μ : } {s₁ : Set α} {s₂ : Set α} {t₁ : Set α} {t₂ : Set α} (hs : s₁ s₂) (hsμ : μ s₂ μ s₁) (ht : t₁ t₂) (htμ : μ t₂ μ t₁) :
μ (s₁ t₁) = μ (s₂ t₂)
@[simp]
theorem MeasureTheory.measure_iUnion_toMeasurable {α : Type u_1} {β : Type u_2} {m : } {μ : } [] (s : βSet α) :
μ (⋃ (b : β), ) = μ (⋃ (b : β), s b)
theorem MeasureTheory.measure_biUnion_toMeasurable {α : Type u_1} {β : Type u_2} {m : } {μ : } {I : Set β} (hc : ) (s : βSet α) :
μ (⋃ (b : β) (_ : b I), ) = μ (⋃ (b : β) (_ : b I), s b)
@[simp]
theorem MeasureTheory.measure_toMeasurable_union {α : Type u_1} {m : } {μ : } {s : Set α} {t : Set α} :
μ () = μ (s t)
@[simp]
theorem MeasureTheory.measure_union_toMeasurable {α : Type u_1} {m : } {μ : } {s : Set α} {t : Set α} :
μ () = μ (s t)
theorem MeasureTheory.sum_measure_le_measure_univ {α : Type u_1} {ι : Type u_5} {m : } {μ : } {s : } {t : ιSet α} (h : ∀ (i : ι), i sMeasurableSet (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 : } {μ : } {s : ιSet 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 : } (μ : ) {s : ιSet 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 : } (μ : ) {s : } {t : ιSet α} (h : ∀ (i : ι), i sMeasurableSet (t i)) (H : μ Set.univ < Finset.sum s fun i => μ (t i)) :
i, i s j, j s _h, Set.Nonempty (t i t j)

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 : } (μ : ) {s : Set α} {t : Set α} {u : Set α} (ht : ) (h's : s u) (h't : t u) (h : μ u < μ s + μ 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 : } (μ : ) {s : Set α} {t : Set α} {u : Set α} (hs : ) (h's : s u) (h't : t u) (h : μ u < μ s + μ 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 : } {μ : } [] {s : ιSet 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 : } {μ : } {s : ιSet α} {t : Set ι} (ht : ) (hd : DirectedOn ((fun x x_1 => x x_1) on s) t) :
μ (⋃ (i : ι) (_ : i t), s i) = ⨆ (i : ι) (_ : i t), μ (s i)
theorem MeasureTheory.measure_iInter_eq_iInf {α : Type u_1} {ι : Type u_5} {m : } {μ : } [] {s : ιSet 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 : } {μ : } [] [] {s : ιSet α} (hm : ) :
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 : } {μ : } [] [] {s : ιSet α} (hs : ∀ (n : ι), MeasurableSet (s n)) (hm : ) (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 : } {μ : } {ι : Type u_8} [] [] [] [] {s : ιSet α} {a : ι} (hs : ∀ (r : ι), r > aMeasurableSet (s r)) (hm : ∀ (i j : ι), a < ii js i s j) (hf : r, r > a μ (s r) ) :
Filter.Tendsto (μ s) (nhdsWithin 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 : } {μ : } {s : Set α} (hs : ∑' (i : ), 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 : } {μ : } {s : Set α} (h : ∑' (i : ), μ (s i) ) :
μ (Filter.liminf s Filter.atTop) = 0
theorem MeasureTheory.limsup_ae_eq_of_forall_ae_eq {α : Type u_1} {m : } {μ : } (s : Set α) {t : Set α} (h : ∀ (n : ), ) :
Filter.limsup s Filter.atTop =ᶠ[] t
theorem MeasureTheory.liminf_ae_eq_of_forall_ae_eq {α : Type u_1} {m : } {μ : } (s : Set α) {t : Set α} (h : ∀ (n : ), ) :
Filter.liminf s Filter.atTop =ᶠ[] t
theorem MeasureTheory.measure_if {α : Type u_1} {β : Type u_2} {m : } {μ : } {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 : ] (m : ) (h : ) :

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 : ] (μ : ) :
@[simp]
theorem MeasureTheory.toMeasure_toOuterMeasure {α : Type u_1} [ms : ] (m : ) (h : ) :
theorem MeasureTheory.toMeasure_apply {α : Type u_1} [ms : ] (m : ) (h : ) {s : Set α} (hs : ) :
s = m s
theorem MeasureTheory.le_toMeasure_apply {α : Type u_1} [ms : ] (m : ) (h : ) (s : Set α) :
m s s
theorem MeasureTheory.toMeasure_apply₀ {α : Type u_1} [ms : ] (m : ) (h : ) {s : Set α} (hs : ) :
s = m s
@[simp]
theorem MeasureTheory.toOuterMeasure_toMeasure {α : Type u_1} [ms : ] {μ : } :
theorem MeasureTheory.boundedBy_measure {α : Type u_1} [ms : ] (μ : ) :
theorem MeasureTheory.Measure.measure_inter_eq_of_measure_eq {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} {u : Set α} (hs : ) (h : μ t = μ u) (htu : t u) (ht_ne_top : μ t ) :
μ (t s) = μ (u s)

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 : } {μ : } {s : Set α} {t : Set α} (hs : ) (ht : μ t ) :
μ () = 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]
theorem MeasureTheory.Measure.zero_toOuterMeasure {α : Type u_1} {_m : } :
0 = 0
@[simp]
theorem MeasureTheory.Measure.coe_zero {α : Type u_1} {_m : } :
0 = 0
instance MeasureTheory.Measure.instSubsingleton {α : Type u_1} [] {m : } :
theorem MeasureTheory.Measure.eq_zero_of_isEmpty {α : Type u_1} [] {_m : } (μ : ) :
μ = 0
instance MeasureTheory.Measure.instAdd {α : Type u_1} [] :
@[simp]
theorem MeasureTheory.Measure.add_toOuterMeasure {α : Type u_1} {_m : } (μ₁ : ) (μ₂ : ) :
↑(μ₁ + μ₂) = μ₁ + μ₂
@[simp]
theorem MeasureTheory.Measure.coe_add {α : Type u_1} {_m : } (μ₁ : ) (μ₂ : ) :
↑(μ₁ + μ₂) = μ₁ + μ₂
theorem MeasureTheory.Measure.add_apply {α : Type u_1} {_m : } (μ₁ : ) (μ₂ : ) (s : Set α) :
↑(μ₁ + μ₂) s = μ₁ s + μ₂ s
instance MeasureTheory.Measure.instSMul {α : Type u_1} {R : Type u_6} [] [] :
@[simp]
theorem MeasureTheory.Measure.smul_toOuterMeasure {α : Type u_1} {R : Type u_6} [] {_m : } (c : R) (μ : ) :
↑(c μ) = c μ
@[simp]
theorem MeasureTheory.Measure.coe_smul {α : Type u_1} {R : Type u_6} [] {_m : } (c : R) (μ : ) :
↑(c μ) = c μ
@[simp]
theorem MeasureTheory.Measure.smul_apply {α : Type u_1} {R : Type u_6} [] {_m : } (c : R) (μ : ) (s : Set α) :
↑(c μ) s = c μ s
instance MeasureTheory.Measure.instSMulCommClass {α : Type u_1} {R : Type u_6} {R' : Type u_7} [] [] [] [] [] :
instance MeasureTheory.Measure.instIsScalarTower {α : Type u_1} {R : Type u_6} {R' : Type u_7} [] [] [] [SMul R R'] [] [] :
instance MeasureTheory.Measure.instIsCentralScalar {α : Type u_1} {R : Type u_6} [] [] [] :
instance MeasureTheory.Measure.instMulAction {α : Type u_1} {R : Type u_6} [] [] [] :
def MeasureTheory.Measure.coeAddHom {α : Type u_1} :
{x : } →

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 : } (I : ) (μ : ) :
↑(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 : } (I : ) (μ : ) (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} [] [] :
instance MeasureTheory.Measure.instModule {α : Type u_1} {R : Type u_6} [] [] [] :
theorem MeasureTheory.Measure.coe_nnreal_smul_apply {α : Type u_1} {_m : } (c : NNReal) (μ : ) (s : Set α) :
↑(c μ) s = c * μ s
@[simp]
theorem MeasureTheory.Measure.nnreal_smul_coe_apply {α : Type u_1} {_m : } (c : NNReal) (μ : ) (s : Set α) :
c μ s = c * μ s
theorem MeasureTheory.Measure.ae_smul_measure_iff {α : Type u_1} {m0 : } {μ : } {p : αProp} {c : ENNReal} (hc : c 0) :
(∀ᵐ (x : α) ∂c μ, p x) ∀ᵐ (x : α) ∂μ, p x
theorem MeasureTheory.Measure.measure_eq_left_of_subset_of_measure_add_eq {α : Type u_1} {m0 : } {μ : } {ν : } {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 : } {μ : } {ν : } {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 : } {μ : } {ν : } {s : Set α} {t : Set α} (hs : ) (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 : } {μ : } {ν : } {s : Set α} {t : Set α} (hs : ) (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 : } {μ₁ : } {μ₂ : } :
μ₁ μ₂ ∀ (s : Set α), μ₁ s μ₂ s
theorem MeasureTheory.Measure.toOuterMeasure_le {α : Type u_1} {m0 : } {μ₁ : } {μ₂ : } :
μ₁ μ₂ μ₁ μ₂
theorem MeasureTheory.Measure.le_iff' {α : Type u_1} {m0 : } {μ₁ : } {μ₂ : } :
μ₁ μ₂ ∀ (s : Set α), μ₁ s μ₂ s
theorem MeasureTheory.Measure.lt_iff {α : Type u_1} {m0 : } {μ : } {ν : } :
μ < ν μ ν s, μ s < ν s
theorem MeasureTheory.Measure.lt_iff' {α : Type u_1} {m0 : } {μ : } {ν : } :
μ < ν μ ν s, μ s < ν s
instance MeasureTheory.Measure.covariantAddLE {α : Type u_1} [] :
CovariantClass () () (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 : } {μ : } {ν : } {ν' : } (h : μ ν) :
μ ν' + ν
theorem MeasureTheory.Measure.le_add_right {α : Type u_1} {m0 : } {μ : } {ν : } {ν' : } (h : μ ν) :
μ ν + ν'
theorem MeasureTheory.Measure.sInf_caratheodory {α : Type u_1} {m0 : } {m : } (s : Set α) (hs : ) :
theorem MeasureTheory.Measure.sInf_apply {α : Type u_1} {m0 : } {s : Set α} {m : } (hs : ) :
↑(sInf m) s = ↑(sInf (MeasureTheory.Measure.toOuterMeasure '' m)) s
@[simp]
@[simp]
theorem MeasureTheory.Measure.top_add {α : Type u_1} {m0 : } {μ : } :
+ μ =
@[simp]
theorem MeasureTheory.Measure.add_top {α : Type u_1} {m0 : } {μ : } :
μ + =
theorem MeasureTheory.Measure.zero_le {α : Type u_1} {_m0 : } (μ : ) :
0 μ
theorem MeasureTheory.Measure.nonpos_iff_eq_zero' {α : Type u_1} {m0 : } {μ : } :
μ 0 μ = 0
@[simp]
theorem MeasureTheory.Measure.measure_univ_eq_zero {α : Type u_1} {m0 : } {μ : } :
μ Set.univ = 0 μ = 0
theorem MeasureTheory.Measure.measure_univ_ne_zero {α : Type u_1} {m0 : } {μ : } :
μ Set.univ 0 μ 0
@[simp]
theorem MeasureTheory.Measure.measure_univ_pos {α : Type u_1} {m0 : } {μ : } :
0 < μ Set.univ μ 0

### Pushforward and pullback #

def MeasureTheory.Measure.liftLinear {α : Type u_1} {β : Type u_2} [] {m0 : } (hf : ∀ (μ : ), inst✝ ) :

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 : } [] {μ : } (hf : ∀ (μ : ), inst✝ ) {s : Set β} (hs : ) :
↑(↑() μ) s = ↑(f μ) s
theorem MeasureTheory.Measure.le_liftLinear_apply {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } (hf : ∀ (μ : ), inst✝ ) (s : Set β) :
↑(f μ) s ↑(↑() μ) s
def MeasureTheory.Measure.mapₗ {α : Type u_1} {β : Type u_2} [] [] (f : αβ) :

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 : } [] {μ : } {f : αβ} {g : αβ} (hf : ) (hg : ) (h : ) :
=
theorem MeasureTheory.Measure.map_def {α : Type u_8} {β : Type u_9} [] [] (f : αβ) (μ : ) :
= if hf : then ↑() μ else 0
@[irreducible]
def MeasureTheory.Measure.map {α : Type u_8} {β : Type u_9} [] [] (f : αβ) (μ : ) :

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 : } [] {μ : } {f : αβ} (hf : ) :
↑() μ =
theorem MeasureTheory.Measure.mapₗ_apply_of_measurable {α : Type u_1} {β : Type u_2} {m0 : } [] {f : αβ} (hf : ) (μ : ) :
@[simp]
theorem MeasureTheory.Measure.map_add {α : Type u_1} {β : Type u_2} {m0 : } [] (μ : ) (ν : ) {f : αβ} (hf : ) :
@[simp]
theorem MeasureTheory.Measure.map_zero {α : Type u_1} {β : Type u_2} {m0 : } [] (f : αβ) :
theorem MeasureTheory.Measure.map_of_not_aemeasurable {α : Type u_1} {β : Type u_2} {m0 : } [] {f : αβ} {μ : } (hf : ) :
theorem MeasureTheory.Measure.map_congr {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {f : αβ} {g : αβ} (h : ) :
@[simp]
theorem MeasureTheory.Measure.map_smul {α : Type u_1} {β : Type u_2} {m0 : } [] (c : ENNReal) (μ : ) (f : αβ) :
@[simp]
theorem MeasureTheory.Measure.map_smul_nnreal {α : Type u_1} {β : Type u_2} {m0 : } [] (c : NNReal) (μ : ) (f : αβ) :
@[simp]
theorem MeasureTheory.Measure.map_apply_of_aemeasurable {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {f : αβ} (hf : ) {s : Set β} (hs : ) :
↑() 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 : } [] {μ : } {f : αβ} (hf : ) {s : Set β} (hs : ) :
↑() s = μ (f ⁻¹' s)
theorem MeasureTheory.Measure.map_toOuterMeasure {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {f : αβ} (hf : ) :
@[simp]
theorem MeasureTheory.Measure.map_id {α : Type u_1} {m0 : } {μ : } :
@[simp]
theorem MeasureTheory.Measure.map_id' {α : Type u_1} {m0 : } {μ : } :
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 : } [] [] {μ : } {g : βγ} {f : αβ} (hg : ) (hf : ) :
theorem MeasureTheory.Measure.map_mono {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {ν : } {f : αβ} (h : μ ν) (hf : ) :
theorem MeasureTheory.Measure.le_map_apply {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {f : αβ} (hf : ) (s : Set β) :
μ (f ⁻¹' s) ↑() 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 : } [] {μ : } {f : αβ} (hf : ) {s : Set β} (hs : ↑() s = 0) :
μ (f ⁻¹' 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 : } [] {μ : } {f : αβ} (hf : ) :
def MeasureTheory.Measure.comapₗ {α : Type u_1} {β : Type u_2} [] [] (f : αβ) :

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} [] {mβ : } (f : αβ) (hfi : ) (hf : ∀ (s : Set α), MeasurableSet (f '' s)) (μ : ) (hs : ) :
↑() s = μ (f '' s)
def MeasureTheory.Measure.comap {α : Type u_1} {β : Type u_2} [] [] (f : αβ) (μ : ) :

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} [] {s : Set α} [] (f : αβ) (μ : ) (hfi : ) (hf : ∀ (s : Set α), ) (hs : ) :
↑() s = μ (f '' s)
theorem MeasureTheory.Measure.le_comap_apply {α : Type u_1} {β : Type u_8} [] {mβ : } (f : αβ) (μ : ) (hfi : ) (hf : ∀ (s : Set α), ) (s : Set α) :
μ (f '' s) ↑() s
theorem MeasureTheory.Measure.comap_apply {α : Type u_1} {s : Set α} {β : Type u_8} [] {_mβ : } (f : αβ) (hfi : ) (hf : ∀ (s : Set α), MeasurableSet (f '' s)) (μ : ) (hs : ) :
↑() s = μ (f '' s)
theorem MeasureTheory.Measure.comapₗ_eq_comap {α : Type u_1} {s : Set α} {β : Type u_8} [] {_mβ : } (f : αβ) (hfi : ) (hf : ∀ (s : Set α), MeasurableSet (f '' s)) (μ : ) (hs : ) :
↑() s = ↑() s
theorem MeasureTheory.Measure.measure_image_eq_zero_of_comap_eq_zero {α : Type u_1} {β : Type u_8} [] {_mβ : } (f : αβ) (μ : ) (hfi : ) (hf : ∀ (s : Set α), ) {s : Set α} (hs : ↑() s = 0) :
μ (f '' s) = 0
theorem MeasureTheory.Measure.ae_eq_image_of_ae_eq_comap {α : Type u_1} {β : Type u_8} [] {mβ : } (f : αβ) (μ : ) (hfi : ) (hf : ∀ (s : Set α), ) {s : Set α} {t : Set α} (hst : ) :
f '' s =ᶠ[] f '' t
theorem MeasureTheory.Measure.NullMeasurableSet.image {α : Type u_1} {β : Type u_8} [] {mβ : } (f : αβ) (μ : ) (hfi : ) (hf : ∀ (s : Set α), ) {s : Set α} (hs : ) :
theorem MeasureTheory.Measure.comap_preimage {α : Type u_1} {β : Type u_8} [] {mβ : } (f : αβ) (μ : ) {s : Set β} (hf : ) (hf' : ) (h : ∀ (t : Set α), ) (hs : ) :
↑() (f ⁻¹' s) = μ (s )

### Subtype of a measure space #

theorem MeasureTheory.Measure.MeasurableSet.nullMeasurableSet_subtype_coe {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set s} (hs : ) (ht : ) :
theorem MeasureTheory.Measure.NullMeasurableSet.subtype_coe {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set s} (hs : ) (ht : ) :
theorem MeasureTheory.Measure.measure_subtype_coe_le_comap {α : Type u_1} {m0 : } {μ : } {s : Set α} (hs : ) (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 : } {μ : } {s : Set α} (hs : ) {t : Set s} (ht : ↑(MeasureTheory.Measure.comap Subtype.val μ) t = 0) :
μ (Subtype.val '' t) = 0
theorem MeasureTheory.Measure.Subtype.volume_def {α : Type u_1} {s : Set α} :
MeasureTheory.volume = MeasureTheory.Measure.comap Subtype.val MeasureTheory.volume
theorem MeasureTheory.Measure.Subtype.volume_univ {α : Type u_1} {s : Set α} (hs : ) :
MeasureTheory.volume Set.univ = MeasureTheory.volume s
theorem MeasureTheory.Measure.volume_subtype_coe_le_volume {α : Type u_1} {s : Set α} (hs : ) (t : Set s) :
MeasureTheory.volume (Subtype.val '' t) MeasureTheory.volume t
theorem MeasureTheory.Measure.volume_subtype_coe_eq_zero_of_volume_eq_zero {α : Type u_1} {s : Set α} (hs : ) {t : Set s} (ht : MeasureTheory.volume t = 0) :
MeasureTheory.volume (Subtype.val '' t) = 0

### Restricting a measure #

def MeasureTheory.Measure.restrictₗ {α : Type u_1} {m0 : } (s : Set α) :

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 : } (μ : ) (s : Set α) :

Restrict a measure μ to a set s.

Instances For
@[simp]
theorem MeasureTheory.Measure.restrictₗ_apply {α : Type u_1} {_m0 : } (s : Set α) (μ : ) :
theorem MeasureTheory.Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict {α : Type u_1} {m0 : } {μ : } {s : Set α} (h : ) :

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 : } {μ : } {s : Set α} {t : Set α} (ht : ) :
t = μ (t s)
@[simp]
theorem MeasureTheory.Measure.restrict_apply {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (ht : ) :
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 : } ⦃s : Set α ⦃s' : Set α ⦃μ : ⦃ν : (hs : ) (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 : } ⦃s : Set α ⦃s' : Set α (hs : s s') ⦃μ : ⦃ν : (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 : } {μ : } {s : Set α} {t : Set α} (h : ) :
theorem MeasureTheory.Measure.restrict_congr_set {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (h : ) :
@[simp]
theorem MeasureTheory.Measure.restrict_apply' {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (hs : ) :
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 : } {μ : } {s : Set α} {t : Set α} (hs : ) :
t = μ (t s)
theorem MeasureTheory.Measure.restrict_le_self {α : Type u_1} {m0 : } {μ : } {s : Set α} :
theorem MeasureTheory.Measure.restrict_eq_self {α : Type u_1} {m0 : } (μ : ) {s : Set α} {t : Set α} (h : s t) :
s = μ s
@[simp]
theorem MeasureTheory.Measure.restrict_apply_self {α : Type u_1} {m0 : } (μ : ) (s : Set α) :
s = μ s
theorem MeasureTheory.Measure.restrict_apply_univ {α : Type u_1} {m0 : } {μ : } (s : Set α) :
Set.univ = μ s
theorem MeasureTheory.Measure.le_restrict_apply {α : Type u_1} {m0 : } {μ : } (s : Set α) (t : Set α) :
μ (t s) t
theorem MeasureTheory.Measure.restrict_apply_superset {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (h : s t) :
t = μ s
@[simp]
theorem MeasureTheory.Measure.restrict_add {α : Type u_1} {_m0 : } (μ : ) (ν : ) (s : Set α) :
@[simp]
theorem MeasureTheory.Measure.restrict_zero {α : Type u_1} {_m0 : } (s : Set α) :
@[simp]
theorem MeasureTheory.Measure.restrict_smul {α : Type u_1} {_m0 : } (c : ENNReal) (μ : ) (s : Set α) :
=
theorem MeasureTheory.Measure.restrict_restrict₀ {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (hs : ) :
@[simp]
theorem MeasureTheory.Measure.restrict_restrict {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (hs : ) :
theorem MeasureTheory.Measure.restrict_restrict_of_subset {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (h : s t) :
theorem MeasureTheory.Measure.restrict_restrict₀' {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (ht : ) :
theorem MeasureTheory.Measure.restrict_restrict' {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (ht : ) :
theorem MeasureTheory.Measure.restrict_comm {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (hs : ) :
theorem MeasureTheory.Measure.restrict_apply_eq_zero {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (ht : ) :
t = 0 μ (t s) = 0
theorem MeasureTheory.Measure.measure_inter_eq_zero_of_restrict {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (h : t = 0) :
μ (t s) = 0
theorem MeasureTheory.Measure.restrict_apply_eq_zero' {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (hs : ) :
t = 0 μ (t s) = 0
@[simp]
theorem MeasureTheory.Measure.restrict_eq_zero {α : Type u_1} {m0 : } {μ : } {s : Set α} :
μ s = 0
instance MeasureTheory.Measure.restrict.neZero {α : Type u_1} {m0 : } {μ : } {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 : } {μ : } {s : Set α} (h : μ s = 0) :
@[simp]
theorem MeasureTheory.Measure.restrict_empty {α : Type u_1} {m0 : } {μ : } :
@[simp]
theorem MeasureTheory.Measure.restrict_univ {α : Type u_1} {m0 : } {μ : } :
theorem MeasureTheory.Measure.restrict_inter_add_diff₀ {α : Type u_1} {m0 : } {μ : } {t : Set α} (s : Set α) (ht : ) :
+ =
theorem MeasureTheory.Measure.restrict_inter_add_diff {α : Type u_1} {m0 : } {μ : } {t : Set α} (s : Set α) (ht : ) :
+ =
theorem MeasureTheory.Measure.restrict_union_add_inter₀ {α : Type u_1} {m0 : } {μ : } {t : Set α} (s : Set α) (ht : ) :
theorem MeasureTheory.Measure.restrict_union_add_inter {α : Type u_1} {m0 : } {μ : } {t : Set α} (s : Set α) (ht : ) :
theorem MeasureTheory.Measure.restrict_union_add_inter' {α : Type u_1} {m0 : } {μ : } {s : Set α} (hs : ) (t : Set α) :
theorem MeasureTheory.Measure.restrict_union₀ {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (h : ) (ht : ) :
theorem MeasureTheory.Measure.restrict_union {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (h : Disjoint s t) (ht : ) :
theorem MeasureTheory.Measure.restrict_union' {α : Type u_1} {m0 : } {μ : } {s : Set α} {t : Set α} (h : Disjoint s t) (hs : ) :
@[simp]
theorem MeasureTheory.Measure.restrict_add_restrict_compl {α : Type u_1} {m0 : } {μ : } {s : Set α} (hs : ) :
@[simp]
theorem MeasureTheory.Measure.restrict_compl_add_restrict {α : Type u_1} {m0 : } {μ : } {s : Set α} (hs : ) :
theorem MeasureTheory.Measure.restrict_union_le {α : Type u_1} {m0 : } {μ : } (s : Set α) (s' : Set α) :
theorem MeasureTheory.Measure.restrict_iUnion_apply_ae {α : Type u_1} {ι : Type u_5} {m0 : } {μ : } [] {s : ιSet α} (hd : ) (hm : ∀ (i : ι), ) {t : Set α} (ht : ) :
↑(MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i)) t = ∑' (i : ι), ↑() t
theorem MeasureTheory.Measure.restrict_iUnion_apply {α : Type u_1} {ι : Type u_5} {m0 : } {μ : } [] {s : ιSet α} (hd : Pairwise (Disjoint on s)) (hm : ∀ (i : ι), MeasurableSet (s i)) {t : Set α} (ht : ) :
↑(MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i)) t = ∑' (i : ι), ↑() t
theorem MeasureTheory.Measure.restrict_iUnion_apply_eq_iSup {α : Type u_1} {ι : Type u_5} {m0 : } {μ : } [] {s : ιSet α} (hd : Directed (fun x x_1 => x x_1) s) {t : Set α} (ht : ) :
↑(MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i)) t = ⨆ (i : ι), ↑() t
theorem MeasureTheory.Measure.restrict_map {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {f : αβ} (hf : ) {s : Set β} (hs : ) :

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 : } {μ : } {s : Set α} (h : μ s ) :
theorem MeasureTheory.Measure.restrict_eq_self_of_ae_mem {α : Type u_1} {_m0 : } ⦃s : Set α ⦃μ : (hs : ∀ᵐ (x : α) ∂μ, x s) :
theorem MeasureTheory.Measure.restrict_congr_meas {α : Type u_1} {m0 : } {μ : } {ν : } {s : Set α} (hs : ) :
∀ (t : Set α), t sμ t = ν t
theorem MeasureTheory.Measure.restrict_congr_mono {α : Type u_1} {m0 : } {μ : } {ν : } {s : Set α} {t : Set α} (hs : s t) :
theorem MeasureTheory.Measure.restrict_union_congr {α : Type u_1} {m0 : } {μ : } {ν : } {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 : } {μ : } {ν : } {s : } {t : ιSet 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 =
theorem MeasureTheory.Measure.restrict_iUnion_congr {α : Type u_1} {ι : Type u_5} {m0 : } {μ : } {ν : } [] {s : ιSet α} :
MeasureTheory.Measure.restrict μ (⋃ (i : ι), s i) = MeasureTheory.Measure.restrict ν (⋃ (i : ι), s i) ∀ (i : ι), =
theorem MeasureTheory.Measure.restrict_biUnion_congr {α : Type u_1} {ι : Type u_5} {m0 : } {μ : } {ν : } {s : Set ι} {t : ιSet α} (hc : ) :
MeasureTheory.Measure.restrict μ (⋃ (i : ι) (_ : i s), t i) = MeasureTheory.Measure.restrict ν (⋃ (i : ι) (_ : i s), t i) ∀ (i : ι), i s =
theorem MeasureTheory.Measure.restrict_sUnion_congr {α : Type u_1} {m0 : } {μ : } {ν : } {S : Set (Set α)} (hc : ) :
∀ (s : Set α),
theorem MeasureTheory.Measure.restrict_sInf_eq_sInf_restrict {α : Type u_1} {t : Set α} {m0 : } {m : } (hm : ) (ht : ) :
= sInf ((fun μ => ) '' 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 : } {μ : } {s : Set α} (hs : μ s 0) {p : αProp} (hp : ∀ᵐ (x : α) ∂, p x) :
x, x s p x

### Extensionality results #

theorem MeasureTheory.Measure.ext_iff_of_iUnion_eq_univ {α : Type u_1} {ι : Type u_5} {m0 : } {μ : } {ν : } [] {s : ιSet α} (hs : ⋃ (i : ι), s i = Set.univ) :
μ = ν ∀ (i : ι), =

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 : } {μ : } {ν : } [] {s : ιSet α} (hs : ⋃ (i : ι), s i = Set.univ) :
(∀ (i : ι), = ) → 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 : } {μ : } {ν : } {S : Set ι} {s : ιSet α} (hc : ) (hs : ⋃ (i : ι) (_ : i S), s i = Set.univ) :
μ = ν ∀ (i : ι), i S =

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 : } {μ : } {ν : } {S : Set ι} {s : ιSet α} (hc : ) (hs : ⋃ (i : ι) (_ : i S), s i = Set.univ) :
(∀ (i : ι), i S = ) → 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 : } {μ : } {ν : } {S : Set (Set α)} (hc : ) (hs : ⋃₀ S = Set.univ) :
μ = ν ∀ (s : Set α),

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 : } {μ : } {ν : } {S : Set (Set α)} (hc : ) (hs : ⋃₀ S = Set.univ) :
(∀ (s : Set α), ) → 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 : } {μ : } {ν : } {S : Set (Set α)} {T : Set (Set α)} (h_gen : ) (hc : ) (h_inter : ) (hU : ⋃₀ T = Set.univ) (htop : ∀ (t : Set α), t Tμ t ) (ST_eq : ∀ (t : Set α), t T∀ (s : Set α), s Sμ (s t) = ν (s t)) (T_eq : ∀ (t : Set α), t Tμ t = ν t) :
μ = ν
theorem MeasureTheory.Measure.ext_of_generateFrom_of_cover_subset {α : Type u_1} {m0 : } {μ : } {ν : } {S : Set (Set α)} {T : Set (Set α)} (h_gen : ) (h_inter : ) (h_sub : T S) (hc : ) (hU : ⋃₀ T = Set.univ) (htop : ∀ (s : Set α), s Tμ s ) (h_eq : ∀ (s : Set α), s Sμ 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 : } {μ : } {ν : } (C : Set (Set α)) (B : Set α) (hA : ) (hC : ) (h1B : ⋃ (i : ), B i = Set.univ) (h2B : ∀ (i : ), B i C) (hμB : ∀ (i : ), μ (B i) ) (h_eq : ∀ (s : Set α), s Cμ 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 : } (f : ) :

Sum of an indexed family of measures.

Instances For
theorem MeasureTheory.Measure.le_sum_apply {α : Type u_1} {ι : Type u_5} {m0 : } (f : ) (s : Set α) :
∑' (i : ι), ↑(f i) s s
@[simp]
theorem MeasureTheory.Measure.sum_apply {α : Type u_1} {ι : Type u_5} {m0 : } (f : ) {s : Set α} (hs : ) :
s = ∑' (i : ι), ↑(f i) s
theorem MeasureTheory.Measure.le_sum {α : Type u_1} {ι : Type u_5} {m0 : } (μ : ) (i : ι) :
@[simp]
theorem MeasureTheory.Measure.sum_apply_eq_zero {α : Type u_1} {ι : Type u_5} {m0 : } [] {μ : } {s : Set α} :
s = 0 ∀ (i : ι), ↑(μ i) s = 0
theorem MeasureTheory.Measure.sum_apply_eq_zero' {α : Type u_1} {ι : Type u_5} {m0 : } {μ : } {s : Set α} (hs : ) :
s = 0 ∀ (i : ι), ↑(μ i) s = 0
theorem MeasureTheory.Measure.sum_comm {α : Type u_1} {ι : Type u_5} {m0 : } {ι' : Type u_8} (μ : ι) :
theorem MeasureTheory.Measure.ae_sum_iff {α : Type u_1} {ι : Type u_5} {m0 : } [] {μ : } {p : αProp} :
(∀ᵐ (x : α) ∂, 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 : } {μ : } {p : αProp} (h : MeasurableSet {x | p x}) :
(∀ᵐ (x : α) ∂, 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 : } [] (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 : } (s : ) (μ : ) :
(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 : } [] (μ : ) :
= ⨆ (i : ι),
theorem MeasureTheory.Measure.sum_bool {α : Type u_1} {m0 : } (f : ) :
theorem MeasureTheory.Measure.sum_cond {α : Type u_1} {m0 : } (μ : ) (ν : ) :
(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 : } (μ : ) {s : Set α} (hs : ) :
theorem MeasureTheory.Measure.sum_of_empty {α : Type u_1} {ι : Type u_5} {m0 : } [] (μ : ) :
theorem MeasureTheory.Measure.sum_add_sum_compl {α : Type u_1} {ι : Type u_5} {m0 : } (s : Set ι) (μ : ) :
((MeasureTheory.Measure.sum fun i => μ i) + MeasureTheory.Measure.sum fun i => μ i) =
theorem MeasureTheory.Measure.sum_congr {α : Type u_1} {m0 : } {μ : } {ν : } (h : ∀ (n : ), μ n = ν n) :
theorem MeasureTheory.Measure.sum_add_sum {α : Type u_1} {m0 : } (μ : ) (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 : } {μ : } [] {s : ιSet α} (hd : ) (hm : ∀ (i : ι), ) :
theorem MeasureTheory.Measure.restrict_iUnion {α : Type u_1} {ι : Type u_5} {m0 : } {μ : } [] {s : ιSet α} (hd : Pairwise (Disjoint on s)) (hm : ∀ (i : ι), MeasurableSet (s i)) :
theorem MeasureTheory.Measure.restrict_iUnion_le {α : Type u_1} {ι : Type u_5} {m0 : } {μ : } [] {s : ιSet α} :

### Absolute continuity #

def MeasureTheory.Measure.AbsolutelyContinuous {α : Type u_1} {_m0 : } (μ : ) (ν : ) :

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 : } {μ : } {ν : } (h : μ ν) :
theorem LE.le.absolutelyContinuous {α : Type u_1} {m0 : } {μ : } {ν : } (h : μ ν) :

Alias of MeasureTheory.Measure.absolutelyContinuous_of_le.

theorem MeasureTheory.Measure.absolutelyContinuous_of_eq {α : Type u_1} {m0 : } {μ : } {ν : } (h : μ = ν) :
theorem Eq.absolutelyContinuous {α : Type u_1} {m0 : } {μ : } {ν : } (h : μ = ν) :

Alias of MeasureTheory.Measure.absolutelyContinuous_of_eq.

theorem MeasureTheory.Measure.AbsolutelyContinuous.mk {α : Type u_1} {m0 : } {μ : } {ν : } (h : ∀ ⦃s : Set α⦄, ν s = 0μ s = 0) :
theorem MeasureTheory.Measure.AbsolutelyContinuous.refl {α : Type u_1} {_m0 : } (μ : ) :
theorem MeasureTheory.Measure.AbsolutelyContinuous.rfl {α : Type u_1} {m0 : } {μ : } :
theorem MeasureTheory.Measure.AbsolutelyContinuous.trans {α : Type u_1} {m0 : } {μ₁ : } {μ₂ : } {μ₃ : } (h1 : ) (h2 : ) :
theorem MeasureTheory.Measure.AbsolutelyContinuous.map {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {ν : } {f : αβ} (hf : ) :
theorem MeasureTheory.Measure.AbsolutelyContinuous.smul {α : Type u_1} {R : Type u_6} {m0 : } {μ : } {ν : } [] (c : R) :
theorem MeasureTheory.Measure.absolutelyContinuous_of_le_smul {α : Type u_1} {m0 : } {μ : } {μ' : } {c : ENNReal} (hμ'_le : μ' c μ) :
theorem MeasureTheory.Measure.ae_le_iff_absolutelyContinuous {α : Type u_1} {m0 : } {μ : } {ν : } :
theorem LE.le.absolutelyContinuous_of_ae {α : Type u_1} {m0 : } {μ : } {ν : } :

Alias of the forward direction of MeasureTheory.Measure.ae_le_iff_absolutelyContinuous.

theorem MeasureTheory.Measure.AbsolutelyContinuous.ae_le {α : Type u_1} {m0 : } {μ : } {ν : } :

Alias of the reverse direction of MeasureTheory.Measure.ae_le_iff_absolutelyContinuous.

theorem MeasureTheory.Measure.ae_mono' {α : Type u_1} {m0 : } {μ : } {ν : } :

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 : } {μ : } {ν : } {f : αδ} {g : αδ} (h' : ) :

### Quasi measure preserving maps (a.k.a. non-singular maps) #

structure MeasureTheory.Measure.QuasiMeasurePreserving {α : Type u_1} {β : Type u_2} [] {m0 : } (f : αβ) (μa : ) (μb : ) :
• measurable :
• absolutelyContinuous :

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 : } (μ : ) :
theorem Measurable.quasiMeasurePreserving {α : Type u_1} {β : Type u_2} [] {f : αβ} {_m0 : } (hf : ) (μ : ) :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.mono_left {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μa' : } {μb : } {f : αβ} (ha : ) :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.mono_right {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μb : } {μb' : } {f : αβ} (ha : ) :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.mono {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μa' : } {μb : } {μb' : } {f : αβ} (ha : ) (hb : ) :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m0 : } [] [] {μa : } {μb : } {μc : } {g : βγ} {f : αβ} :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.iterate {α : Type u_1} {m0 : } {μa : } {f : αα} (n : ) :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.aemeasurable {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μb : } {f : αβ} :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.ae_map_le {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μb : } {f : αβ} :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.tendsto_ae {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μb : } {f : αβ} :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.ae {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μb : } {f : αβ} {p : βProp} (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 : } [] {μa : } {μb : } {f : αβ} {g₁ : βδ} {g₂ : βδ} (hg : g₁ =ᶠ[] g₂) :
g₁ f =ᶠ[] g₂ f
theorem MeasureTheory.Measure.QuasiMeasurePreserving.preimage_null {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μb : } {f : αβ} {s : Set β} (hs : μb s = 0) :
μa (f ⁻¹' s) = 0
theorem MeasureTheory.Measure.QuasiMeasurePreserving.preimage_mono_ae {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μb : } {f : αβ} {s : Set β} {t : Set β} (h : ) :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.preimage_ae_eq {α : Type u_1} {β : Type u_2} {m0 : } [] {μa : } {μb : } {f : αβ} {s : Set β} {t : Set β} (h : ) :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.preimage_iterate_ae_eq {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : αα} (k : ) (hs : ) :
theorem MeasureTheory.Measure.QuasiMeasurePreserving.image_zpow_ae_eq {α : Type u_1} {m0 : } {μ : } {s : Set α} {e : α α} (he' : ) (k : ) (hs : e '' s =ᶠ[] s) :
↑(e ^ k) '' s =ᶠ[] s
theorem MeasureTheory.Measure.QuasiMeasurePreserving.limsup_preimage_iterate_ae_eq {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : αα} (hs : ) :
Filter.limsup (fun n => )^[n] s) Filter.atTop =ᶠ[] s
theorem MeasureTheory.Measure.QuasiMeasurePreserving.liminf_preimage_iterate_ae_eq {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : αα} (hs : ) :
Filter.liminf (fun n => )^[n] s) Filter.atTop =ᶠ[] s
theorem MeasureTheory.Measure.QuasiMeasurePreserving.exists_preimage_eq_of_preimage_ae {α : Type u_1} {m0 : } {μ : } {s : Set α} {f : αα} (hs : ) (hs' : ) :
t, 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} [] [] [] {s : Set α} {t : Set α} {μ : } (g : G) (h_qmp : MeasureTheory.Measure.QuasiMeasurePreserving ((fun x x_1 => x +ᵥ x_1) (-g))) (h_ae_eq : ) :
g +ᵥ s =ᶠ[] g +ᵥ t
theorem MeasureTheory.Measure.QuasiMeasurePreserving.smul_ae_eq_of_ae_eq {G : Type u_8} {α : Type u_9} [] [] [] {s : Set α} {t : Set α} {μ : } (g : G) (h_qmp : MeasureTheory.Measure.QuasiMeasurePreserving ((fun x x_1 => x x_1) g⁻¹)) (h_ae_eq : ) :
g s =ᶠ[] g t
theorem MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_zero {G : Type u_8} {α : Type u_9} [] [] [] {μ : } {s : Set α} (h_ae_disjoint : ∀ (g : G), g 0MeasureTheory.AEDisjoint μ (g +ᵥ s) s) (h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving ((fun x x_1 => x +ᵥ x_1) g)) :
Pairwise ( on fun g => g +ᵥ s)
theorem MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_one {G : Type u_8} {α : Type u_9} [] [] [] {μ : } {s : Set α} (h_ae_disjoint : ∀ (g : G), g 1MeasureTheory.AEDisjoint μ (g s) s) (h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving ((fun x x_1 => x x_1) g)) :
Pairwise ( on fun g => g s)

### The cofinite filter #

def MeasureTheory.Measure.cofinite {α : Type u_1} {m0 : } (μ : ) :

The filter of sets s such that sᶜ has finite measure.

Instances For
theorem MeasureTheory.Measure.mem_cofinite {α : Type u_1} {m0 : } {μ : } {s : Set α} :
μ s <
theorem MeasureTheory.Measure.compl_mem_cofinite {α : Type u_1} {m0 : } {μ : } {s : Set α} :
μ s <
theorem MeasureTheory.Measure.eventually_cofinite {α : Type u_1} {m0 : } {μ : } {p : αProp} :
(∀ᶠ (x : α) in , p x) μ {x | ¬p x} <
theorem MeasureTheory.NullMeasurableSet.preimage {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {ν : } {f : αβ} {t : Set β} (ht : ) :

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 : } {μ : } {ν : } {s : Set α} (hle : ) :
theorem MeasureTheory.NullMeasurableSet.mono {α : Type u_1} {m0 : } {μ : } {ν : } {s : Set α} (hle : ν μ) :
theorem MeasureTheory.AEDisjoint.preimage {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {ν : } {f : αβ} {s : Set β} {t : Set β} (ht : ) :
@[simp]
theorem MeasureTheory.ae_eq_bot {α : Type u_1} {m0 : } {μ : } :
μ = 0
@[simp]
theorem MeasureTheory.ae_neBot {α : Type u_1} {m0 : } {μ : } :
instance MeasureTheory.Measure.ae.neBot {α : Type u_1} {m0 : } {μ : } [] :
@[simp]
theorem MeasureTheory.ae_zero {α : Type u_1} {_m0 : } :
theorem MeasureTheory.ae_mono {α : Type u_1} {m0 : } {μ : } {ν : } (h : μ ν) :
theorem MeasureTheory.mem_ae_map_iff {α : Type u_1} {β : Type u_2} {m0 : } [] {μ : } {f : αβ} (hf : ) {s : Set β} (hs : ) :
theorem MeasureTheory.mem_ae_of_mem_ae_map {α : Type u_1} {β : Type u_2} {m0 : } [] {μ :