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∞.

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.

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 : Measure α} : (ae μ).IsMeasurablyGenerated

theorem MeasureTheory.ae_uIoc_iff {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} [LinearOrder α] {a b : α} {P : α → Prop} : (∀ᵐ (x : α) ∂μ, x ∈ Set.uIoc a b → P x) ↔ (∀ᵐ (x : α) ∂μ, x ∈ Set.Ioc a b → P x) ∧ ∀ᵐ (x : α) ∂μ, x ∈ Set.Ioc b a → P x

See also MeasureTheory.ae_restrict_uIoc_iff.

theorem MeasureTheory.measure_union {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s₁ s₂ : Set α} (hd : Disjoint s₁ s₂) (h : MeasurableSet s₂) : μ (s₁ ∪ s₂) = μ s₁ + μ s₂

theorem MeasureTheory.measure_union' {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s₁ s₂ : Set α} (hd : Disjoint s₁ s₂) (h : MeasurableSet s₁) : μ (s₁ ∪ s₂) = μ s₁ + μ s₂

theorem MeasureTheory.measure_inter_add_diff {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {t : Set α} (s : Set α) (ht : MeasurableSet t) : μ (s ∩ t) + μ (s \ t) = μ s

theorem MeasureTheory.measure_diff_add_inter {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {t : Set α} (s : Set α) (ht : MeasurableSet t) : μ (s \ t) + μ (s ∩ t) = μ s

theorem MeasureTheory.measure_diff_eq_top {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (hs : μ s = ⊤) (ht : μ t ≠ ⊤) : μ (s \ t) = ⊤

theorem MeasureTheory.measure_union_add_inter {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {t : Set α} (s : Set α) (ht : MeasurableSet t) : μ (s ∪ t) + μ (s ∩ t) = μ s + μ t

theorem MeasureTheory.measure_union_add_inter' {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s : Set α} (hs : MeasurableSet s) (t : Set α) : μ (s ∪ t) + μ (s ∩ t) = μ s + μ t

theorem MeasureTheory.measure_symmDiff_eq {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) : μ (symmDiff s t) = μ (s \ t) + μ (t \ s)

theorem MeasureTheory.measure_symmDiff_le {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} (s t u : Set α) : μ (symmDiff s u) ≤ μ (symmDiff s t) + μ (symmDiff t u)

theorem MeasureTheory.measure_symmDiff_eq_top {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (hs : μ s ≠ ⊤) (ht : μ t = ⊤) : μ (symmDiff s t) = ⊤

theorem MeasureTheory.measure_add_measure_compl {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s : Set α} (h : MeasurableSet s) : μ s + μ sᶜ = μ Set.univ

theorem MeasureTheory.measure_biUnion₀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α} {s : Set β} {f : β → Set α} (hs : s.Countable) (hd : s.Pairwise (Function.onFun (AEDisjoint μ) f)) (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : μ (⋃ b ∈ s, f b) = ∑' (p : ↑s), μ (f ↑p)

theorem MeasureTheory.measure_biUnion {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α} {s : Set β} {f : β → Set α} (hs : s.Countable) (hd : s.PairwiseDisjoint f) (h : ∀ b ∈ s, MeasurableSet (f b)) : μ (⋃ b ∈ s, f b) = ∑' (p : ↑s), μ (f ↑p)

theorem MeasureTheory.measure_sUnion₀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise (AEDisjoint μ)) (h : ∀ s ∈ S, NullMeasurableSet s μ) : μ (⋃₀ S) = ∑' (s : ↑S), μ ↑s

theorem MeasureTheory.measure_sUnion {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise Disjoint) (h : ∀ s ∈ S, MeasurableSet s) : μ (⋃₀ S) = ∑' (s : ↑S), μ ↑s

theorem MeasureTheory.measure_biUnion_finset₀ {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : Measure α} {s : Finset ι} {f : ι → Set α} (hd : (↑s).Pairwise (Function.onFun (AEDisjoint μ) f)) (hm : ∀ b ∈ s, NullMeasurableSet (f b) μ) : μ (⋃ b ∈ s, f b) = ∑ p ∈ s, μ (f p)

theorem MeasureTheory.measure_biUnion_finset {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : Measure α} {s : Finset ι} {f : ι → Set α} (hd : (↑s).PairwiseDisjoint f) (hm : ∀ b ∈ s, MeasurableSet (f b)) : μ (⋃ b ∈ s, f b) = ∑ p ∈ s, μ (f p)

theorem MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint₀ {α : Type u_1} {ι : Type u_8} {x✝ : MeasurableSpace α} (μ : Measure α) {As : ι → Set α} (As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ) (As_disj : Pairwise (Function.onFun (AEDisjoint μ) 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} {x✝ : MeasurableSpace α} (μ : Measure α) {As : ι → Set α} (As_mble : ∀ (i : ι), MeasurableSet (As i)) (As_disj : Pairwise (Function.onFun Disjoint 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 : Measure α} {s : Set β} (hs : s.Countable) {f : α → β} (hf : ∀ y ∈ s, MeasurableSet (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.measure_preimage_eq_zero_iff_of_countable {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α} {s : Set β} {f : α → β} (hs : s.Countable) : μ (f ⁻¹' s) = 0 ↔ ∀ x ∈ s, μ (f ⁻¹' {x}) = 0

theorem MeasureTheory.sum_measure_preimage_singleton {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α} (s : Finset β) {f : α → β} (hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})) : ∑ b ∈ s, 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}.

@[simp]

theorem MeasureTheory.sum_measure_singleton {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s : Finset α} [MeasurableSingletonClass α] : ∑ x ∈ s, μ {x} = μ ↑s

theorem MeasureTheory.measure_diff_null' {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s₁ s₂ : Set α} (h : μ (s₁ ∩ s₂) = 0) : μ (s₁ \ s₂) = μ s₁

theorem MeasureTheory.measure_add_diff {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s : Set α} (hs : NullMeasurableSet s μ) (t : Set α) : μ s + μ (t \ s) = μ (s ∪ t)

theorem MeasureTheory.measure_diff' {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {t : Set α} (s : Set α) (hm : NullMeasurableSet t μ) (h_fin : μ t ≠ ⊤) : μ (s \ t) = μ (s ∪ t) - μ t

theorem MeasureTheory.measure_diff {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s₁ s₂ : Set α} (h : s₂ ⊆ s₁) (h₂ : NullMeasurableSet s₂ μ) (h_fin : μ s₂ ≠ ⊤) : μ (s₁ \ s₂) = μ s₁ - μ s₂

theorem MeasureTheory.le_measure_diff {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s₁ s₂ : Set α} : μ s₁ - μ s₂ ≤ μ (s₁ \ s₂)

theorem MeasureTheory.measure_eq_top_iff_of_symmDiff {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (hμst : μ (symmDiff s t) ≠ ⊤) : μ s = ⊤ ↔ μ t = ⊤

If the measure of the symmetric difference of two sets is finite, then one has infinite measure if and only if the other one does.

theorem MeasureTheory.measure_ne_top_iff_of_symmDiff {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (hμst : μ (symmDiff s t) ≠ ⊤) : μ s ≠ ⊤ ↔ μ t ≠ ⊤

If the measure of the symmetric difference of two sets is finite, then one has finite measure if and only if the other one does.

theorem MeasureTheory.measure_diff_lt_of_lt_add {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (hs : NullMeasurableSet s μ) (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 : MeasurableSpace α} {μ : Measure α} {s t : Set α} (hs : NullMeasurableSet s μ) (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 : MeasurableSpace α} {μ : Measure α} {s 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 : Measure α} {s₁ s₂ 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 : MeasurableSpace α} {μ : Measure α} {s₁ s₂ 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 : Measure α} {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂) (h23 : s₂ ⊆ s₃) (h_nulldiff : μ (s₃ \ s₁) = 0) : μ s₂ = μ s₃

theorem MeasureTheory.measure_compl₀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s : Set α} (h : NullMeasurableSet s μ) (hs : μ s ≠ ⊤) : μ sᶜ = μ Set.univ - μ s

theorem MeasureTheory.measure_compl {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s : Set α} (h₁ : MeasurableSet s) (h_fin : μ s ≠ ⊤) : μ sᶜ = μ Set.univ - μ s

theorem MeasureTheory.measure_inter_conull' {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (ht : μ (s \ t) = 0) : μ (s ∩ t) = μ s

theorem MeasureTheory.measure_inter_conull {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (ht : μ tᶜ = 0) : μ (s ∩ t) = μ s

@[simp]

theorem MeasureTheory.union_ae_eq_left_iff_ae_subset {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} : s ∪ t =ᶠ[ae μ] s ↔ t ≤ᶠ[ae μ] s

@[simp]

theorem MeasureTheory.union_ae_eq_right_iff_ae_subset {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} : s ∪ t =ᶠ[ae μ] t ↔ s ≤ᶠ[ae μ] t

theorem MeasureTheory.ae_eq_of_ae_subset_of_measure_ge {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (h₁ : s ≤ᶠ[ae μ] t) (h₂ : μ t ≤ μ s) (hsm : NullMeasurableSet s μ) (ht : μ t ≠ ⊤) : s =ᶠ[ae μ] t

theorem MeasureTheory.ae_eq_of_subset_of_measure_ge {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} (h₁ : s ⊆ t) (h₂ : μ t ≤ μ s) (hsm : NullMeasurableSet s μ) (ht : μ t ≠ ⊤) : s =ᶠ[ae 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} {m : MeasurableSpace α} {μ : Measure α} {ι : Sort u_8} [Countable ι] {s t : ι → Set α} (hsub : ∀ (i : ι), s i ⊆ t i) (h_le : ∀ (i : ι), μ (t i) ≤ μ (s i)) : μ (⋃ (i : ι), s i) = μ (⋃ (i : ι), t i)

theorem MeasureTheory.measure_union_congr_of_subset {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s₁ s₂ t₁ 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} {m : MeasurableSpace α} {μ : Measure α} {ι : Sort u_8} [Countable ι] (s : ι → Set α) : μ (⋃ (i : ι), toMeasurable μ (s i)) = μ (⋃ (i : ι), s i)

theorem MeasureTheory.measure_biUnion_toMeasurable {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α} {I : Set β} (hc : I.Countable) (s : β → Set α) : μ (⋃ b ∈ I, toMeasurable μ (s b)) = μ (⋃ b ∈ I, s b)

@[simp]

theorem MeasureTheory.measure_toMeasurable_union {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} : μ (toMeasurable μ s ∪ t) = μ (s ∪ t)

@[simp]

theorem MeasureTheory.measure_union_toMeasurable {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {s t : Set α} : μ (s ∪ 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 : Measure α} {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, NullMeasurableSet (t i) μ) (H : (↑s).Pairwise (Function.onFun (AEDisjoint μ) t)) : ∑ i ∈ s, 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 : Measure α} {s : ι → Set α} (hs : ∀ (i : ι), NullMeasurableSet (s i) μ) (H : Pairwise (Function.onFun (AEDisjoint μ) 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 : Measure α) {s : ι → Set α} (hs : ∀ (i : ι), NullMeasurableSet (s i) μ) (H : μ Set.univ < ∑' (i : ι), μ (s i)) : ∃ (i : ι) (j : ι), i ≠ j ∧ (s i ∩ s j).Nonempty

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 : Measure α) {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, NullMeasurableSet (t i) μ) (H : μ Set.univ < ∑ i ∈ s, μ (t i)) : ∃ i ∈ s, ∃ j ∈ s, ∃ (_ : i ≠ j), (t i ∩ t j).Nonempty

Pigeonhole principle for measure spaces: if s is a Finset and ∑ i ∈ 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 : Measure α) {s t u : Set α} (ht : MeasurableSet t) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t) : (s ∩ t).Nonempty

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 : Measure α) {s t u : Set α} (hs : MeasurableSet s) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t) : (s ∩ t).Nonempty

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 Directed.measure_iUnion {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hd : Directed (fun (x1 x2 : Set α) => x1 ⊆ x2) 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 Monotone.measure_iUnion {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Preorder ι] [IsDirected ι fun (x1 x2 : ι) => x1 ≤ x2] [Filter.atTop.IsCountablyGenerated] {s : ι → Set α} (hs : Monotone s) : μ (⋃ (i : ι), s i) = ⨆ (i : ι), μ (s i)

Continuity from below: the measure of the union of a monotone family of sets is equal to the supremum of their measures. The theorem assumes that the atTop filter on the index set is countably generated, so it works for a family indexed by a countable type, as well as ℝ.

theorem Antitone.measure_iUnion {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Preorder ι] [IsDirected ι fun (x1 x2 : ι) => x1 ≥ x2] [Filter.atBot.IsCountablyGenerated] {s : ι → Set α} (hs : Antitone s) : μ (⋃ (i : ι), s i) = ⨆ (i : ι), μ (s i)

theorem MeasureTheory.measure_iUnion_eq_iSup_accumulate {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : Measure α} [Preorder ι] [IsDirected ι fun (x1 x2 : ι) => x1 ≤ x2] [Filter.atTop.IsCountablyGenerated] {f : ι → Set α} : μ (⋃ (i : ι), f i) = ⨆ (i : ι), μ (Set.Accumulate f i)

Continuity from below: the measure of the union of a sequence of (not necessarily measurable) sets is the supremum of the measures of the partial unions.

theorem MeasureTheory.measure_biUnion_eq_iSup {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : Measure α} {s : ι → Set α} {t : Set ι} (ht : t.Countable) (hd : DirectedOn (Function.onFun (fun (x1 x2 : Set α) => x1 ⊆ x2) s) t) : μ (⋃ i ∈ t, s i) = ⨆ i ∈ t, μ (s i)

theorem Directed.measure_iInter {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (h : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) (hd : Directed (fun (x1 x2 : Set α) => x1 ⊇ x2) s) (hfin : ∃ (i : ι), μ (s i) ≠ ⊤) : μ (⋂ (i : ι), s i) = ⨅ (i : ι), μ (s i)

Continuity from above: the measure of the intersection of a directed downwards countable family of measurable sets is the infimum of the measures.

theorem Monotone.measure_iInter {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Preorder ι] [IsDirected ι fun (x1 x2 : ι) => x1 ≥ x2] [Filter.atBot.IsCountablyGenerated] {s : ι → Set α} (hs : Monotone s) (hsm : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) (hfin : ∃ (i : ι), μ (s i) ≠ ⊤) : μ (⋂ (i : ι), s i) = ⨅ (i : ι), μ (s i)

Continuity from above: the measure of the intersection of a monotone family of measurable sets indexed by a type with countably generated atBot filter is equal to the infimum of the measures.

theorem Antitone.measure_iInter {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Preorder ι] [IsDirected ι fun (x1 x2 : ι) => x1 ≤ x2] [Filter.atTop.IsCountablyGenerated] {s : ι → Set α} (hs : Antitone s) (hsm : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) (hfin : ∃ (i : ι), μ (s i) ≠ ⊤) : μ (⋂ (i : ι), s i) = ⨅ (i : ι), μ (s i)

Continuity from above: the measure of the intersection of an antitone family of measurable sets indexed by a type with countably generated atTop filter is equal to the infimum of the measures.

theorem MeasureTheory.measure_iInter_eq_iInf_measure_iInter_le {α : Type u_8} {ι : Type u_9} {x✝ : MeasurableSpace α} {μ : Measure α} [Countable ι] [Preorder ι] [IsDirected ι fun (x1 x2 : ι) => x1 ≤ x2] {f : ι → Set α} (h : ∀ (i : ι), NullMeasurableSet (f i) μ) (hfin : ∃ (i : ι), μ (f i) ≠ ⊤) : μ (⋂ (i : ι), f i) = ⨅ (i : ι), μ (⋂ (j : ι), ⋂ (_ : j ≤ i), f j)

Continuity from above: the measure of the intersection of a sequence of measurable sets is the infimum of the measures of the partial intersections.

theorem MeasureTheory.tendsto_measure_iUnion_atTop {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : Measure α} [Preorder ι] [Filter.atTop.IsCountablyGenerated] {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 (not necessarily measurable) sets is the limit of the measures.

theorem MeasureTheory.tendsto_measure_iUnion_atBot {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : Measure α} [Preorder ι] [Filter.atBot.IsCountablyGenerated] {s : ι → Set α} (hm : Antitone s) : Filter.Tendsto (⇑μ ∘ s) Filter.atBot (nhds (μ (⋃ (n : ι), s n)))

theorem MeasureTheory.tendsto_measure_iUnion_accumulate {α : Type u_8} {ι : Type u_9} [Preorder ι] [Filter.atTop.IsCountablyGenerated] {x✝ : MeasurableSpace α} {μ : Measure α} {f : ι → Set α} : Filter.Tendsto (fun (i : ι) => μ (Set.Accumulate f i)) Filter.atTop (nhds (μ (⋃ (i : ι), f i)))

Continuity from below: the measure of the union of a sequence of (not necessarily measurable) sets is the limit of the measures of the partial unions.

theorem MeasureTheory.tendsto_measure_iInter_atTop {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : Measure α} [Preorder ι] [Filter.atTop.IsCountablyGenerated] {s : ι → Set α} (hs : ∀ (i : ι), NullMeasurableSet (s i) μ) (hm : Antitone s) (hf : ∃ (i : ι), 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_iInter_atBot {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : Measure α} [Preorder ι] [Filter.atBot.IsCountablyGenerated] {s : ι → Set α} (hs : ∀ (i : ι), NullMeasurableSet (s i) μ) (hm : Monotone s) (hf : ∃ (i : ι), μ (s i) ≠ ⊤) : Filter.Tendsto (⇑μ ∘ s) Filter.atBot (nhds (μ (⋂ (n : ι), s n)))

Continuity from above: the measure of the intersection of an increasing sequence of measurable sets is the limit of the measures.

theorem MeasureTheory.tendsto_measure_iInter_le {α : Type u_8} {ι : Type u_9} {x✝ : MeasurableSpace α} {μ : Measure α} [Countable ι] [Preorder ι] {f : ι → Set α} (hm : ∀ (i : ι), NullMeasurableSet (f i) μ) (hf : ∃ (i : ι), μ (f i) ≠ ⊤) : Filter.Tendsto (fun (i : ι) => μ (⋂ (j : ι), ⋂ (_ : j ≤ i), f j)) Filter.atTop (nhds (μ (⋂ (i : ι), f i)))

Continuity from above: the measure of the intersection of a sequence of measurable sets such that one has finite measure is the limit of the measures of the partial intersections.

theorem MeasureTheory.exists_measure_iInter_lt {α : Type u_8} {ι : Type u_9} {x✝ : MeasurableSpace α} {μ : Measure α} [SemilatticeSup ι] [Countable ι] {f : ι → Set α} (hm : ∀ (i : ι), NullMeasurableSet (f i) μ) {ε : ENNReal} (hε : 0 < ε) (hfin : ∃ (i : ι), μ (f i) ≠ ⊤) (hfem : ⋂ (n : ι), f n = ∅) : ∃ (m : ι), μ (⋂ (n : ι), ⋂ (_ : n ≤ m), f n) < ε

Some version of continuity of a measure in the empty set using the intersection along a set of sets.

theorem MeasureTheory.tendsto_measure_biInter_gt {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {ι : Type u_8} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α} {a : ι} (hs : ∀ r > a, NullMeasurableSet (s r) μ) (hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j) (hf : ∃ 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_if {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α} {x : β} {t : Set β} {s : Set α} [Decidable (x ∈ t)] : μ (if x ∈ t then s else ∅) = t.indicator (fun (x : β) => μ s) x

def MeasureTheory.OuterMeasure.toMeasure {α : Type u_1} [ms : MeasurableSpace α] (m : OuterMeasure α) (h : ms ≤ m.caratheodory) : Measure α

Obtain a measure by giving an outer measure where all sets in the σ-algebra are Carathéodory measurable.

Equations

• m.toMeasure h = MeasureTheory.Measure.ofMeasurable (fun (s : Set α) (x : MeasurableSet s) => m s) ⋯ ⋯

theorem MeasureTheory.le_toOuterMeasure_caratheodory {α : Type u_1} [ms : MeasurableSpace α] (μ : Measure α) : ms ≤ μ.caratheodory

@[simp]

theorem MeasureTheory.toMeasure_toOuterMeasure {α : Type u_1} [ms : MeasurableSpace α] (m : OuterMeasure α) (h : ms ≤ m.caratheodory) : (m.toMeasure h).toOuterMeasure = m.trim

@[simp]

theorem MeasureTheory.toMeasure_apply {α : Type u_1} [ms : MeasurableSpace α] (m : OuterMeasure α) (h : ms ≤ m.caratheodory) {s : Set α} (hs : MeasurableSet s) : (m.toMeasure h) s = m s

theorem MeasureTheory.le_toMeasure_apply {α : Type u_1} [ms : MeasurableSpace α] (m : OuterMeasure α) (h : ms ≤ m.caratheodory) (s : Set α) : m s ≤ (m.toMeasure h) s

theorem MeasureTheory.toMeasure_apply₀ {α : Type u_1} [ms : MeasurableSpace α] (m : OuterMeasure α) (h : ms ≤ m.caratheodory) {s : Set α} (hs : NullMeasurableSet s (m.toMeasure h)) : (m.toMeasure h) s = m s

@[simp]

theorem MeasureTheory.toOuterMeasure_toMeasure {α : Type u_1} [ms : MeasurableSpace α] {μ : Measure α} : μ.toMeasure ⋯ = μ

@[simp]

theorem MeasureTheory.boundedBy_measure {α : Type u_1} [ms : MeasurableSpace α] (μ : Measure α) : OuterMeasure.boundedBy ⇑μ = μ.toOuterMeasure

theorem MeasureTheory.Measure.measure_inter_eq_of_measure_eq {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} {s t u : Set α} (hs : MeasurableSet s) (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 : MeasurableSpace α} {μ : Measure α} {s t : Set α} (hs : MeasurableSet s) (ht : μ t ≠ ⊤) : μ (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 s-finite (for example when it is σ-finite), see measure_toMeasurable_inter_of_sFinite.

The ℝ≥0∞-module of measures

instance MeasureTheory.Measure.instZero {α : Type u_1} {x✝ : MeasurableSpace α} : Zero (Measure α)

Equations

• MeasureTheory.Measure.instZero = { zero := let __OuterMeasure := 0; { toOuterMeasure := __OuterMeasure, m_iUnion := ⋯, trim_le := ⋯ } }

@[simp]

theorem MeasureTheory.Measure.zero_toOuterMeasure {α : Type u_1} {_m : MeasurableSpace α} : toOuterMeasure 0 = 0

@[simp]

theorem MeasureTheory.Measure.coe_zero {α : Type u_1} {_m : MeasurableSpace α} : ⇑0 = 0

@[simp]

theorem MeasureTheory.OuterMeasure.toMeasure_zero {α : Type u_1} [ms : MeasurableSpace α] (h : ms ≤ OuterMeasure.caratheodory 0) : toMeasure 0 h = 0

@[simp]

theorem MeasureTheory.OuterMeasure.toMeasure_eq_zero {α : Type u_1} {ms : MeasurableSpace α} {μ : OuterMeasure α} (h : ms ≤ μ.caratheodory) : μ.toMeasure h = 0 ↔ μ = 0

theorem MeasureTheory.Measure.apply_eq_zero_of_isEmpty {α : Type u_1} [IsEmpty α] {x✝ : MeasurableSpace α} (μ : Measure α) (s : Set α) : μ s = 0

instance MeasureTheory.Measure.instSubsingleton {α : Type u_1} [IsEmpty α] {m : MeasurableSpace α} : Subsingleton (Measure α)

theorem MeasureTheory.Measure.eq_zero_of_isEmpty {α : Type u_1} [IsEmpty α] {_m : MeasurableSpace α} (μ : Measure α) : μ = 0

instance MeasureTheory.Measure.instInhabited {α : Type u_1} {x✝ : MeasurableSpace α} : Inhabited (Measure α)

Equations

• MeasureTheory.Measure.instInhabited = { default := 0 }

instance MeasureTheory.Measure.instAdd {α : Type u_1} {x✝ : MeasurableSpace α} : Add (Measure α)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem MeasureTheory.Measure.add_toOuterMeasure {α : Type u_1} {_m : MeasurableSpace α} (μ₁ μ₂ : Measure α) : (μ₁ + μ₂).toOuterMeasure = μ₁.toOuterMeasure + μ₂.toOuterMeasure

@[simp]

theorem MeasureTheory.Measure.coe_add {α : Type u_1} {_m : MeasurableSpace α} (μ₁ μ₂ : Measure α) : ⇑(μ₁ + μ₂) = ⇑μ₁ + ⇑μ₂

theorem MeasureTheory.Measure.add_apply {α : Type u_1} {_m : MeasurableSpace α} (μ₁ μ₂ : Measure α) (s : Set α) : (μ₁ + μ₂) s = μ₁ s + μ₂ s

instance MeasureTheory.Measure.instSMul {α : Type u_1} {R : Type u_6} [SMul R ENNReal] [IsScalarTower R ENNReal ENNReal] {x✝ : MeasurableSpace α} : SMul R (Measure α)

Equations

• One or more equations did not get rendered due to their size.

@[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 : Measure α) : (c • μ).toOuterMeasure = c • μ.toOuterMeasure

@[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 : Measure α) : ⇑(c • μ) = c • ⇑μ

@[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 : Measure α) (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} [SMul R ENNReal] [IsScalarTower R ENNReal ENNReal] [SMul R' ENNReal] [IsScalarTower R' ENNReal ENNReal] [SMulCommClass R R' ENNReal] {x✝ : MeasurableSpace α} : SMulCommClass R R' (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] {x✝ : MeasurableSpace α} : IsScalarTower R R' (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] {x✝ : MeasurableSpace α} : IsCentralScalar R (Measure α)

instance MeasureTheory.Measure.instNoZeroSMulDivisors {α : Type u_1} {R : Type u_6} {m0 : MeasurableSpace α} [Zero R] [SMulWithZero R ENNReal] [IsScalarTower R ENNReal ENNReal] [NoZeroSMulDivisors R ENNReal] : NoZeroSMulDivisors R (Measure α)

instance MeasureTheory.Measure.instMulAction {α : Type u_1} {R : Type u_6} [Monoid R] [MulAction R ENNReal] [IsScalarTower R ENNReal ENNReal] {x✝ : MeasurableSpace α} : MulAction R (Measure α)

Equations

• MeasureTheory.Measure.instMulAction = Function.Injective.mulAction MeasureTheory.Measure.toOuterMeasure ⋯ ⋯

instance MeasureTheory.Measure.instAddCommMonoid {α : Type u_1} {x✝ : MeasurableSpace α} : AddCommMonoid (Measure α)

Equations

• MeasureTheory.Measure.instAddCommMonoid = Function.Injective.addCommMonoid MeasureTheory.Measure.toOuterMeasure ⋯ ⋯ ⋯ ⋯

def MeasureTheory.Measure.coeAddHom {α : Type u_1} {x✝ : MeasurableSpace α} : Measure α →+ Set α → ENNReal

Coercion to function as an additive monoid homomorphism.

Equations

• MeasureTheory.Measure.coeAddHom = { toFun := DFunLike.coe, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem MeasureTheory.Measure.coeAddHom_apply {α : Type u_1} {x✝ : MeasurableSpace α} (μ : Measure α) : coeAddHom μ = ⇑μ

@[simp]

theorem MeasureTheory.Measure.coe_finset_sum {α : Type u_1} {ι : Type u_5} {_m : MeasurableSpace α} (I : Finset ι) (μ : ι → Measure α) : ⇑(∑ i ∈ I, μ i) = ∑ i ∈ 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 → Measure α) (s : Set α) : (∑ i ∈ I, μ i) s = ∑ i ∈ 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] {x✝ : MeasurableSpace α} : DistribMulAction R (Measure α)

Equations

• MeasureTheory.Measure.instDistribMulAction = Function.Injective.distribMulAction { toFun := MeasureTheory.Measure.toOuterMeasure, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯

instance MeasureTheory.Measure.instModule {α : Type u_1} {R : Type u_6} [Semiring R] [Module R ENNReal] [IsScalarTower R ENNReal ENNReal] {x✝ : MeasurableSpace α} : Module R (Measure α)

Equations

• MeasureTheory.Measure.instModule = Function.Injective.module R { toFun := MeasureTheory.Measure.toOuterMeasure, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯

@[simp]

theorem MeasureTheory.Measure.coe_nnreal_smul_apply {α : Type u_1} {_m : MeasurableSpace α} (c : NNReal) (μ : Measure α) (s : Set α) : (c • μ) s = ↑c * μ s

@[simp]

theorem MeasureTheory.Measure.nnreal_smul_coe_apply {α : Type u_1} {_m : MeasurableSpace α} (c : NNReal) (μ : Measure α) (s : Set α) : c • μ s = ↑c * μ s

theorem MeasureTheory.Measure.ae_smul_measure {α : Type u_1} {R : Type u_6} {m0 : MeasurableSpace α} {μ : Measure α} {p : α → Prop} [SMul R ENNReal] [IsScalarTower R ENNReal ENNReal] (h : ∀ᵐ (x : α) ∂μ, p x) (c : R) : ∀ᵐ (x : α) ∂c • μ, p x

theorem MeasureTheory.Measure.ae_smul_measure_le {α : Type u_1} {R : Type u_6} {m0 : MeasurableSpace α} {μ : Measure α} [SMul R ENNReal] [IsScalarTower R ENNReal ENNReal] (c : R) : ae (c • μ) ≤ ae μ

theorem MeasureTheory.Measure.ae_smul_measure_iff {α : Type u_1} {m0 : MeasurableSpace α} {R : Type u_8} [Zero R] [SMulWithZero R ENNReal] [IsScalarTower R ENNReal ENNReal] [NoZeroSMulDivisors R ENNReal] {c : R} {p : α → Prop} (hc : c ≠ 0) {μ : Measure α} : (∀ᵐ (x : α) ∂c • μ, p x) ↔ ∀ᵐ (x : α) ∂μ, p x

@[simp]

theorem MeasureTheory.Measure.ae_smul_measure_eq {α : Type u_1} {m0 : MeasurableSpace α} {R : Type u_8} [Zero R] [SMulWithZero R ENNReal] [IsScalarTower R ENNReal ENNReal] [NoZeroSMulDivisors R ENNReal] {c : R} (hc : c ≠ 0) (μ : Measure α) : ae (c • μ) = ae μ

theorem MeasureTheory.Measure.measure_eq_left_of_subset_of_measure_add_eq {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : Measure α} {s 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 ν : Measure α} {s 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 ν : Measure α} {s t : Set α} (hs : MeasurableSet s) (ht : (μ + ν) t ≠ ⊤) : μ (toMeasurable (μ + ν) t ∩ s) = μ (t ∩ s)

theorem MeasureTheory.Measure.measure_toMeasurable_add_inter_right {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : Measure α} {s t : Set α} (hs : MeasurableSet s) (ht : (μ + ν) t ≠ ⊤) : ν (toMeasurable (μ + ν) t ∩ s) = ν (t ∩ s)

The complete lattice of measures

instance MeasureTheory.Measure.instPartialOrder {α : Type u_1} {x✝ : MeasurableSpace α} : PartialOrder (Measure α)

Measures are partially ordered.

Equations

• MeasureTheory.Measure.instPartialOrder = { le := fun (m₁ m₂ : MeasureTheory.Measure α) => ∀ (s : Set α), m₁ s ≤ m₂ s, le_refl := ⋯, le_trans := ⋯, lt_iff_le_not_le := ⋯, le_antisymm := ⋯ }

theorem MeasureTheory.Measure.toOuterMeasure_le {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ : Measure α} : μ₁.toOuterMeasure ≤ μ₂.toOuterMeasure ↔ μ₁ ≤ μ₂

theorem MeasureTheory.Measure.le_iff {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ : Measure α} : μ₁ ≤ μ₂ ↔ ∀ (s : Set α), MeasurableSet s → μ₁ s ≤ μ₂ s

theorem MeasureTheory.Measure.le_intro {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ : Measure α} (h : ∀ (s : Set α), MeasurableSet s → s.Nonempty → μ₁ s ≤ μ₂ s) : μ₁ ≤ μ₂

theorem MeasureTheory.Measure.le_iff' {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ : Measure α} : μ₁ ≤ μ₂ ↔ ∀ (s : Set α), μ₁ s ≤ μ₂ s

theorem MeasureTheory.Measure.lt_iff {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : Measure α} : μ < ν ↔ μ ≤ ν ∧ ∃ (s : Set α), MeasurableSet s ∧ μ s < ν s

theorem MeasureTheory.Measure.lt_iff' {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : Measure α} : μ < ν ↔ μ ≤ ν ∧ ∃ (s : Set α), μ s < ν s

instance MeasureTheory.Measure.instAddLeftMono {α : Type u_1} {x✝ : MeasurableSpace α} : AddLeftMono (Measure α)

theorem MeasureTheory.Measure.le_add_left {α : Type u_1} {m0 : MeasurableSpace α} {μ ν ν' : Measure α} (h : μ ≤ ν) : μ ≤ ν' + ν

theorem MeasureTheory.Measure.le_add_right {α : Type u_1} {m0 : MeasurableSpace α} {μ ν ν' : Measure α} (h : μ ≤ ν) : μ ≤ ν + ν'

theorem MeasureTheory.Measure.sInf_caratheodory {α : Type u_1} {m0 : MeasurableSpace α} {m : Set (Measure α)} (s : Set α) (hs : MeasurableSet s) : MeasurableSet s

instance MeasureTheory.Measure.instInfSet {α : Type u_1} {x✝ : MeasurableSpace α} : InfSet (Measure α)

Equations

• MeasureTheory.Measure.instInfSet = { sInf := fun (m : Set (MeasureTheory.Measure α)) => (sInf (MeasureTheory.Measure.toOuterMeasure '' m)).toMeasure ⋯ }

theorem MeasureTheory.Measure.sInf_apply {α : Type u_1} {m0 : MeasurableSpace α} {s : Set α} {m : Set (Measure α)} (hs : MeasurableSet s) : (sInf m) s = (sInf (toOuterMeasure '' m)) s

instance MeasureTheory.Measure.instCompleteSemilatticeInf {α : Type u_1} {x✝ : MeasurableSpace α} : CompleteSemilatticeInf (Measure α)

Equations

• MeasureTheory.Measure.instCompleteSemilatticeInf = { toPartialOrder := inferInstance, toInfSet := inferInstance, sInf_le := ⋯, le_sInf := ⋯ }

instance MeasureTheory.Measure.instCompleteLattice {α : Type u_1} {x✝ : MeasurableSpace α} : CompleteLattice (Measure α)

Equations

• One or more equations did not get rendered due to their size.

theorem MeasureTheory.Measure.inf_apply {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : Measure α} {s : Set α} (hs : MeasurableSet s) : (μ ⊓ ν) s = sInf {m : ENNReal | ∃ (t : Set α), m = μ (t ∩ s) + ν (tᶜ ∩ s)}

@[simp]

theorem MeasureTheory.OuterMeasure.toMeasure_top {α : Type u_1} {m0 : MeasurableSpace α} : ⊤.toMeasure ⋯ = ⊤

@[simp]

theorem MeasureTheory.Measure.toOuterMeasure_top {α : Type u_1} {x✝ : MeasurableSpace α} : ⊤.toOuterMeasure = ⊤

@[simp]

theorem MeasureTheory.Measure.top_add {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} : ⊤ + μ = ⊤

@[simp]

theorem MeasureTheory.Measure.add_top {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} : μ + ⊤ = ⊤

theorem MeasureTheory.Measure.zero_le {α : Type u_1} {_m0 : MeasurableSpace α} (μ : Measure α) : 0 ≤ μ

theorem MeasureTheory.Measure.nonpos_iff_eq_zero' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} : μ ≤ 0 ↔ μ = 0

@[simp]

theorem MeasureTheory.Measure.measure_univ_eq_zero {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} : μ Set.univ = 0 ↔ μ = 0

theorem MeasureTheory.Measure.measure_univ_ne_zero {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} : μ Set.univ ≠ 0 ↔ μ ≠ 0

instance MeasureTheory.Measure.instNeZeroENNRealCoeSetUniv {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [NeZero μ] : NeZero (μ Set.univ)

@[simp]

theorem MeasureTheory.Measure.measure_univ_pos {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} : 0 < μ Set.univ ↔ μ ≠ 0

theorem MeasureTheory.Measure.nonempty_of_neZero {α : Type u_1} {m0 : MeasurableSpace α} (μ : Measure α) [NeZero μ] : Nonempty α

noncomputable def MeasureTheory.Measure.sum {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (f : ι → Measure α) : Measure α

Sum of an indexed family of measures.

Equations

• MeasureTheory.Measure.sum f = (MeasureTheory.OuterMeasure.sum fun (i : ι) => (f i).toOuterMeasure).toMeasure ⋯

theorem MeasureTheory.Measure.le_sum_apply {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (f : ι → Measure α) (s : Set α) : ∑' (i : ι), (f i) s ≤ (sum f) s

@[simp]

theorem MeasureTheory.Measure.sum_apply {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (f : ι → Measure α) {s : Set α} (hs : MeasurableSet s) : (sum f) s = ∑' (i : ι), (f i) s

theorem MeasureTheory.Measure.sum_apply₀ {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (f : ι → Measure α) {s : Set α} (hs : NullMeasurableSet s (sum f)) : (sum f) s = ∑' (i : ι), (f i) s

For the next theorem, the countability assumption is necessary. For a counterexample, consider an uncountable space, with a distinguished point x₀, and the sigma-algebra made of countable sets not containing x₀, and their complements. All points but x₀ are measurable. Consider the sum of the Dirac masses at points different from x₀, and s = {x₀}. For any Dirac mass δ_x, we have δ_x (x₀) = 0, so ∑' x, δ_x (x₀) = 0. On the other hand, the measure sum δ_x gives mass one to each point different from x₀, so it gives infinite mass to any measurable set containing x₀ (as such a set is uncountable), and by outer regularity one gets sum δ_x {x₀} = ∞.

theorem MeasureTheory.Measure.sum_apply_of_countable {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [Countable ι] (f : ι → Measure α) (s : Set α) : (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 → Measure α) (i : ι) : μ i ≤ 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 → Measure α} {s : Set α} : (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 → Measure α} {s : Set α} (hs : MeasurableSet s) : (sum μ) s = 0 ↔ ∀ (i : ι), (μ i) s = 0

@[simp]

theorem MeasureTheory.Measure.sum_eq_zero {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} {f : ι → Measure α} : sum f = 0 ↔ ∀ (i : ι), f i = 0

@[simp]

theorem MeasureTheory.Measure.sum_zero {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} : (sum fun (x : ι) => 0) = 0

theorem MeasureTheory.Measure.sum_sum {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} {ι' : Type u_8} (μ : ι → ι' → Measure α) : (sum fun (n : ι) => sum (μ n)) = sum fun (p : ι × ι') => μ p.1 p.2

theorem MeasureTheory.Measure.sum_comm {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} {ι' : Type u_8} (μ : ι → ι' → Measure α) : (sum fun (n : ι) => sum (μ n)) = sum fun (m : ι') => sum fun (n : ι) => μ n m

theorem MeasureTheory.Measure.ae_sum_iff {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [Countable ι] {μ : ι → Measure α} {p : α → Prop} : (∀ᵐ (x : α) ∂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 → Measure α} {p : α → Prop} (h : MeasurableSet {x : α | p x}) : (∀ᵐ (x : α) ∂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 → Measure α) : sum μ = ∑ i : ι, μ i

theorem MeasureTheory.Measure.sum_coe_finset {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (s : Finset ι) (μ : ι → Measure α) : (sum fun (i : { x : ι // x ∈ s }) => μ ↑i) = ∑ i ∈ s, μ i

@[simp]

theorem MeasureTheory.Measure.ae_sum_eq {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [Countable ι] (μ : ι → Measure α) : ae (sum μ) = ⨆ (i : ι), ae (μ i)

theorem MeasureTheory.Measure.sum_bool {α : Type u_1} {m0 : MeasurableSpace α} (f : Bool → Measure α) : sum f = f true + f false

theorem MeasureTheory.Measure.sum_cond {α : Type u_1} {m0 : MeasurableSpace α} (μ ν : Measure α) : (sum fun (b : Bool) => bif b then μ else ν) = μ + ν

@[simp]

theorem MeasureTheory.Measure.sum_of_isEmpty {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [IsEmpty ι] (μ : ι → Measure α) : sum μ = 0

theorem MeasureTheory.Measure.sum_add_sum_compl {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (s : Set ι) (μ : ι → Measure α) : ((sum fun (i : ↑s) => μ ↑i) + sum fun (i : ↑sᶜ) => μ ↑i) = sum μ

theorem MeasureTheory.Measure.sum_congr {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : ℕ → Measure α} (h : ∀ (n : ℕ), μ n = ν n) : sum μ = sum ν

theorem MeasureTheory.Measure.sum_add_sum {α : Type u_1} {m0 : MeasurableSpace α} {ι : Type u_8} (μ ν : ι → Measure α) : sum μ + sum ν = sum fun (n : ι) => μ n + ν n

@[simp]

theorem MeasureTheory.Measure.sum_comp_equiv {α : Type u_1} {m0 : MeasurableSpace α} {ι : Type u_8} {ι' : Type u_9} (e : ι' ≃ ι) (m : ι → Measure α) : sum (m ∘ ⇑e) = sum m

@[simp]

theorem MeasureTheory.Measure.sum_extend_zero {α : Type u_1} {m0 : MeasurableSpace α} {ι : Type u_8} {ι' : Type u_9} {f : ι → ι'} (hf : Function.Injective f) (m : ι → Measure α) : sum (Function.extend f m 0) = sum m

The cofinite filter

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

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

Equations

• μ.cofinite = Filter.comk (fun (x : Set α) => μ x < ⊤) ⋯ ⋯ ⋯

theorem MeasureTheory.Measure.mem_cofinite {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} {s : Set α} : s ∈ μ.cofinite ↔ μ sᶜ < ⊤

theorem MeasureTheory.Measure.compl_mem_cofinite {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} {s : Set α} : sᶜ ∈ μ.cofinite ↔ μ s < ⊤

theorem MeasureTheory.Measure.eventually_cofinite {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} {p : α → Prop} : (∀ᶠ (x : α) in μ.cofinite, p x) ↔ μ {x : α | ¬p x} < ⊤

instance MeasureTheory.Measure.cofinite.instIsMeasurablyGenerated {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} : μ.cofinite.IsMeasurablyGenerated

theorem AEMeasurable.nullMeasurable {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {f : α → β} (h : AEMeasurable f μ) : MeasureTheory.NullMeasurable f μ

theorem AEMeasurable.nullMeasurableSet_preimage {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {f : α → β} {s : Set β} (hf : AEMeasurable f μ) (hs : MeasurableSet s) : MeasureTheory.NullMeasurableSet (f ⁻¹' s) μ

@[simp]

theorem MeasureTheory.ae_eq_bot {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} : ae μ = ⊥ ↔ μ = 0

@[simp]

theorem MeasureTheory.ae_neBot {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} : (ae μ).NeBot ↔ μ ≠ 0

instance MeasureTheory.Measure.ae.neBot {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [NeZero μ] : (ae μ).NeBot

@[simp]

theorem MeasureTheory.ae_zero {α : Type u_1} {_m0 : MeasurableSpace α} : ae 0 = ⊥

theorem MeasureTheory.biSup_measure_Iic {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [Preorder α] {s : Set α} (hsc : s.Countable) (hst : ∀ (x : α), ∃ y ∈ s, x ≤ y) (hdir : DirectedOn (fun (x1 x2 : α) => x1 ≤ x2) s) : ⨆ x ∈ s, μ (Set.Iic x) = μ Set.univ

theorem MeasureTheory.tendsto_measure_Ico_atTop {α : Type u_1} {m0 : MeasurableSpace α} [Preorder α] [NoMaxOrder α] [Filter.atTop.IsCountablyGenerated] (μ : Measure α) (a : α) : Filter.Tendsto (fun (x : α) => μ (Set.Ico a x)) Filter.atTop (nhds (μ (Set.Ici a)))

theorem MeasureTheory.tendsto_measure_Ioc_atBot {α : Type u_1} {m0 : MeasurableSpace α} [Preorder α] [NoMinOrder α] [Filter.atBot.IsCountablyGenerated] (μ : Measure α) (a : α) : Filter.Tendsto (fun (x : α) => μ (Set.Ioc x a)) Filter.atBot (nhds (μ (Set.Iic a)))

theorem MeasureTheory.tendsto_measure_Iic_atTop {α : Type u_1} {m0 : MeasurableSpace α} [Preorder α] [Filter.atTop.IsCountablyGenerated] (μ : Measure α) : Filter.Tendsto (fun (x : α) => μ (Set.Iic x)) Filter.atTop (nhds (μ Set.univ))

theorem MeasureTheory.tendsto_measure_Ici_atBot {α : Type u_1} {m0 : MeasurableSpace α} [Preorder α] [Filter.atBot.IsCountablyGenerated] (μ : Measure α) : Filter.Tendsto (fun (x : α) => μ (Set.Ici x)) Filter.atBot (nhds (μ Set.univ))

theorem MeasureTheory.Iio_ae_eq_Iic' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [PartialOrder α] {a : α} (ha : μ {a} = 0) : Set.Iio a =ᶠ[ae μ] Set.Iic a

theorem MeasureTheory.Ioi_ae_eq_Ici' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [PartialOrder α] {a : α} (ha : μ {a} = 0) : Set.Ioi a =ᶠ[ae μ] Set.Ici a

theorem MeasureTheory.Ioo_ae_eq_Ioc' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [PartialOrder α] {a b : α} (hb : μ {b} = 0) : Set.Ioo a b =ᶠ[ae μ] Set.Ioc a b

theorem MeasureTheory.Ioc_ae_eq_Icc' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [PartialOrder α] {a b : α} (ha : μ {a} = 0) : Set.Ioc a b =ᶠ[ae μ] Set.Icc a b

theorem MeasureTheory.Ioo_ae_eq_Ico' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [PartialOrder α] {a b : α} (ha : μ {a} = 0) : Set.Ioo a b =ᶠ[ae μ] Set.Ico a b

theorem MeasureTheory.Ioo_ae_eq_Icc' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [PartialOrder α] {a b : α} (ha : μ {a} = 0) (hb : μ {b} = 0) : Set.Ioo a b =ᶠ[ae μ] Set.Icc a b

theorem MeasureTheory.Ico_ae_eq_Icc' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [PartialOrder α] {a b : α} (hb : μ {b} = 0) : Set.Ico a b =ᶠ[ae μ] Set.Icc a b

theorem MeasureTheory.Ico_ae_eq_Ioc' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} [PartialOrder α] {a b : α} (ha : μ {a} = 0) (hb : μ {b} = 0) : Set.Ico a b =ᶠ[ae μ] Set.Ioc a b