mathlib documentation

measure_theory.measure_space

Measure spaces #

The definition of a measure and a measure space are in measure_theory.measure_space_def, 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 measure_space_def, and to be available in measure_space (through measurable_space).

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:

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 #

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:

To prove that two measures are equal, there are multiple options:

A measure_space is a class that is a measurable space with a canonical measure. The measure is denoted volume.

References #

Tags #

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

theorem measure_theory.measure_Union {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} [encodable β] {f : β → set α} (hn : pairwise (disjoint on f)) (h : ∀ (i : β), measurable_set (f i)) :
μ (⋃ (i : β), f i) = ∑' (i : β), μ (f i)
theorem measure_theory.measure_union {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s₁ s₂ : set α} (hd : disjoint s₁ s₂) (h₁ : measurable_set s₁) (h₂ : measurable_set s₂) :
μ (s₁ s₂) = μ s₁ + μ s₂
theorem measure_theory.measure_add_measure_compl {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (h : measurable_set s) :
μ s + μ s = μ set.univ
theorem measure_theory.measure_bUnion {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set β} {f : β → set α} (hs : s.countable) (hd : s.pairwise_on (disjoint on f)) (h : ∀ (b : β), b smeasurable_set (f b)) :
μ (⋃ (b : β) (H : b s), f b) = ∑' (p : s), μ (f p)
theorem measure_theory.measure_sUnion {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {S : set (set α)} (hs : S.countable) (hd : S.pairwise_on disjoint) (h : ∀ (s : set α), s Smeasurable_set s) :
μ (⋃₀S) = ∑' (s : S), μ s
theorem measure_theory.measure_bUnion_finset {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} {s : finset ι} {f : ι → set α} (hd : s.pairwise_on (disjoint on f)) (hm : ∀ (b : ι), b smeasurable_set (f b)) :
μ (⋃ (b : ι) (H : b s), f b) = ∑ (p : ι) in s, μ (f p)
theorem measure_theory.tsum_measure_preimage_singleton {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set β} (hs : s.countable) {f : α → β} (hf : ∀ (y : β), y smeasurable_set (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 measure_theory.sum_measure_preimage_singleton {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} (s : finset β) {f : α → β} (hf : ∀ (y : β), y smeasurable_set (f ⁻¹' {y})) :
∑ (b : β) in 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}.

theorem measure_theory.measure_diff_null' {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s₁ s₂ : set α} (h : μ (s₁ s₂) = 0) :
μ (s₁ \ s₂) = μ s₁
theorem measure_theory.measure_diff_null {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s₁ s₂ : set α} (h : μ s₂ = 0) :
μ (s₁ \ s₂) = μ s₁
theorem measure_theory.measure_diff {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s₁ s₂ : set α} (h : s₂ s₁) (h₁ : measurable_set s₁) (h₂ : measurable_set s₂) (h_fin : μ s₂ < ) :
μ (s₁ \ s₂) = μ s₁ - μ s₂
theorem measure_theory.meas_eq_meas_of_null_diff {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (hst : s t) (h_nulldiff : μ (t.diff s) = 0) :
μ s = μ t
theorem measure_theory.meas_eq_meas_of_between_null_diff {α : Type u_1} [measurable_space α] {μ : measure_theory.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 measure_theory.meas_eq_meas_smaller_of_between_null_diff {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s₁ s₂ s₃ : set α} (h12 : s₁ s₂) (h23 : s₂ s₃) (h_nulldiff : μ (s₃.diff s₁) = 0) :
μ s₁ = μ s₂
theorem measure_theory.meas_eq_meas_larger_of_between_null_diff {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s₁ s₂ s₃ : set α} (h12 : s₁ s₂) (h23 : s₂ s₃) (h_nulldiff : μ (s₃.diff s₁) = 0) :
μ s₂ = μ s₃
theorem measure_theory.measure_compl {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (h₁ : measurable_set s) (h_fin : μ s < ) :
μ s = μ set.univ - μ s
theorem measure_theory.sum_measure_le_measure_univ {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} {s : finset ι} {t : ι → set α} (h : ∀ (i : ι), i smeasurable_set (t i)) (H : s.pairwise_on (disjoint on t)) :
∑ (i : ι) in s, μ (t i) μ set.univ
theorem measure_theory.tsum_measure_le_measure_univ {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} {s : ι → set α} (hs : ∀ (i : ι), measurable_set (s i)) (H : pairwise (disjoint on s)) :
∑' (i : ι), μ (s i) μ set.univ
theorem measure_theory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure {α : Type u_1} {ι : Type u_5} [measurable_space α] (μ : measure_theory.measure α) {s : ι → set α} (hs : ∀ (i : ι), measurable_set (s i)) (H : μ set.univ < ∑' (i : ι), μ (s i)) :
∃ (i j : ι) (h : 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 measure_theory.exists_nonempty_inter_of_measure_univ_lt_sum_measure {α : Type u_1} {ι : Type u_5} [measurable_space α] (μ : measure_theory.measure α) {s : finset ι} {t : ι → set α} (h : ∀ (i : ι), i smeasurable_set (t i)) (H : μ set.univ < ∑ (i : ι) in s, μ (t i)) :
∃ (i : ι) (H : i s) (j : ι) (H : j s) (h : i j), (t i t j).nonempty

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 measure_theory.measure_Union_eq_supr {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} [encodable ι] {s : ι → set α} (h : ∀ (i : ι), measurable_set (s i)) (hd : directed has_subset.subset s) :
μ (⋃ (i : ι), s i) = ⨆ (i : ι), μ (s i)

Continuity from below: the measure of the union of a directed sequence of measurable sets is the supremum of the measures.

theorem measure_theory.measure_bUnion_eq_supr {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} {s : ι → set α} {t : set ι} (ht : t.countable) (h : ∀ (i : ι), i tmeasurable_set (s i)) (hd : directed_on (has_subset.subset on s) t) :
μ (⋃ (i : ι) (H : i t), s i) = ⨆ (i : ι) (H : i t), μ (s i)
theorem measure_theory.measure_Inter_eq_infi {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} [encodable ι] {s : ι → set α} (h : ∀ (i : ι), measurable_set (s i)) (hd : directed superset 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 decreasing sequence of measurable sets is the infimum of the measures.

theorem measure_theory.measure_eq_inter_diff {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (ht : measurable_set t) :
μ s = μ (s t) + μ (s \ t)
theorem measure_theory.measure_union_add_inter {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (ht : measurable_set t) :
μ (s t) + μ (s t) = μ s + μ t
theorem measure_theory.tendsto_measure_Union {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : ∀ (n : ), measurable_set (s n)) (hm : monotone s) :
filter.tendsto (μ s) filter.at_top (𝓝 (μ (⋃ (n : ), s n)))

Continuity from below: the measure of the union of an increasing sequence of measurable sets is the limit of the measures.

theorem measure_theory.tendsto_measure_Inter {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : ∀ (n : ), measurable_set (s n)) (hm : ∀ ⦃n m : ⦄, n ms m s n) (hf : ∃ (i : ), μ (s i) < ) :
filter.tendsto (μ s) filter.at_top (𝓝 (μ (⋂ (n : ), s n)))

Continuity from above: the measure of the intersection of a decreasing sequence of measurable sets is the limit of the measures.

theorem measure_theory.measure_limsup_eq_zero {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : ∀ (i : ), measurable_set (s i)) (hs' : ∑' (i : ), μ (s i) ) :

One direction of the Borel-Cantelli lemma: if (sᵢ) is a sequence of measurable sets such that ∑ μ sᵢ exists, then the limit superior of the sᵢ is a null set.

theorem measure_theory.measure_if {α : Type u_1} {β : Type u_2} [measurable_space α] {x : β} {t : set β} {s : set α} {μ : measure_theory.measure α} :
μ (ite (x t) s ) = t.indicator (λ (_x : β), μ s) x

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

Equations
@[simp]
theorem measure_theory.to_measure_apply {α : Type u_1} [ms : measurable_space α] (m : measure_theory.outer_measure α) (h : ms m.caratheodory) {s : set α} (hs : measurable_set s) :
(m.to_measure h) s = m s
theorem measure_theory.le_to_measure_apply {α : Type u_1} [ms : measurable_space α] (m : measure_theory.outer_measure α) (h : ms m.caratheodory) (s : set α) :
m s (m.to_measure h) s
theorem measure_theory.measure.caratheodory {α : Type u_1} [measurable_space α] {s t : set α} (μ : measure_theory.measure α) (hs : measurable_set s) :
μ (t s) + μ (t \ s) = μ t

The ℝ≥0∞-module of measures #

@[simp]
theorem measure_theory.measure.coe_zero {α : Type u_1} [measurable_space α] :
0 = 0
@[simp]
@[simp]
theorem measure_theory.measure.coe_add {α : Type u_1} [measurable_space α] (μ₁ μ₂ : measure_theory.measure α) :
(μ₁ + μ₂) = μ₁ + μ₂
theorem measure_theory.measure.add_apply {α : Type u_1} [measurable_space α] (μ₁ μ₂ : measure_theory.measure α) (s : set α) :
(μ₁ + μ₂) s = μ₁ s + μ₂ s
@[simp]
theorem measure_theory.measure.coe_smul {α : Type u_1} [measurable_space α] (c : ℝ≥0∞) (μ : measure_theory.measure α) :
(c μ) = c μ
theorem measure_theory.measure.smul_apply {α : Type u_1} [measurable_space α] (c : ℝ≥0∞) (μ : measure_theory.measure α) (s : set α) :
(c μ) s = c * μ s
@[simp]
theorem measure_theory.measure.coe_nnreal_smul {α : Type u_1} [measurable_space α] (c : ℝ≥0) (μ : measure_theory.measure α) :
(c μ) = c μ

The complete lattice of measures #

@[instance]

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.

Equations
theorem measure_theory.measure.le_iff {α : Type u_1} [measurable_space α] {μ₁ μ₂ : measure_theory.measure α} :
μ₁ μ₂ ∀ (s : set α), measurable_set sμ₁ s μ₂ s
theorem measure_theory.measure.to_outer_measure_le {α : Type u_1} [measurable_space α] {μ₁ μ₂ : measure_theory.measure α} :
μ₁.to_outer_measure μ₂.to_outer_measure μ₁ μ₂
theorem measure_theory.measure.le_iff' {α : Type u_1} [measurable_space α] {μ₁ μ₂ : measure_theory.measure α} :
μ₁ μ₂ ∀ (s : set α), μ₁ s μ₂ s
theorem measure_theory.measure.lt_iff {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} :
μ < ν μ ν ∃ (s : set α), measurable_set s μ s < ν s
theorem measure_theory.measure.lt_iff' {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} :
μ < ν μ ν ∃ (s : set α), μ s < ν s
theorem measure_theory.measure.le_add_left {α : Type u_1} [measurable_space α] {μ ν ν' : measure_theory.measure α} (h : μ ν) :
μ ν' + ν
theorem measure_theory.measure.le_add_right {α : Type u_1} [measurable_space α] {μ ν ν' : measure_theory.measure α} (h : μ ν) :
μ ν + ν'
theorem measure_theory.measure.zero_le {α : Type u_1} [measurable_space α] (μ : measure_theory.measure α) :
0 μ
@[simp]

Pushforward and pullback #

Lift a linear map between outer_measure 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.

Equations

The pushforward of a measure. It is defined to be 0 if f is not a measurable function.

Equations
@[simp]
theorem measure_theory.measure.map_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : measurable f) {s : set β} (hs : measurable_set s) :

We can evaluate the pushforward on measurable sets. For non-measurable sets, see measure_theory.measure.le_map_apply and measurable_equiv.map_apply.

theorem measure_theory.measure.map_of_not_measurable {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : ¬measurable f) :
theorem measure_theory.measure.map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] [measurable_space β] [measurable_space γ] {μ : measure_theory.measure α} {g : β → γ} {f : α → β} (hg : measurable g) (hf : measurable f) :
theorem measure_theory.measure.map_mono {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ ν : measure_theory.measure α} (f : α → β) (h : μ ν) :
theorem measure_theory.measure.le_map_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : measurable f) (s : set β) :

Even if s is not measurable, we can bound map f μ s from below. See also measurable_equiv.map_apply.

theorem measure_theory.measure.preimage_null_of_map_null {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : measurable f) {s : set β} (hs : ((measure_theory.measure.map f) μ) 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 measure_theory.measure.tendsto_ae_map {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : measurable f) :

Pullback of a measure. If f sends each measurable set to a measurable set, then for each measurable set s we have comap f μ s = μ (f '' s).

Equations
theorem measure_theory.measure.comap_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {s : set α} (f : α → β) (hfi : function.injective f) (hf : ∀ (s : set α), measurable_set smeasurable_set (f '' s)) (μ : measure_theory.measure β) (hs : measurable_set s) :

Restricting a measure #

Restrict a measure μ to a set s.

Equations
@[simp]
theorem measure_theory.measure.restrict_apply {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (ht : measurable_set t) :
(μ.restrict s) t = μ (t s)

If t is a measurable set, then the measure of t with respect to the restriction of the measure to s equals the outer measure of t ∩ s. An alternate version requiring that s be measurable instead of t exists as measure.restrict_apply'.

theorem measure_theory.measure.restrict_eq_self {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (h_meas_t : measurable_set t) (h : t s) :
(μ.restrict s) t = μ t
theorem measure_theory.measure.restrict_apply_self {α : Type u_1} [measurable_space α] {s : set α} (μ : measure_theory.measure α) (h_meas_s : measurable_set s) :
(μ.restrict s) s = μ s
theorem measure_theory.measure.le_restrict_apply {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} (s t : set α) :
μ (t s) (μ.restrict s) t
@[simp]
theorem measure_theory.measure.restrict_add {α : Type u_1} [measurable_space α] (μ ν : measure_theory.measure α) (s : set α) :
+ ν).restrict s = μ.restrict s + ν.restrict s
@[simp]
theorem measure_theory.measure.restrict_zero {α : Type u_1} [measurable_space α] (s : set α) :
0.restrict s = 0
@[simp]
theorem measure_theory.measure.restrict_smul {α : Type u_1} [measurable_space α] (c : ℝ≥0∞) (μ : measure_theory.measure α) (s : set α) :
(c μ).restrict s = c μ.restrict s
@[simp]
theorem measure_theory.measure.restrict_restrict {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) :
(μ.restrict t).restrict s = μ.restrict (s t)
theorem measure_theory.measure.restrict_comm {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (ht : measurable_set t) :
(μ.restrict t).restrict s = (μ.restrict s).restrict t
theorem measure_theory.measure.restrict_apply_eq_zero {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (ht : measurable_set t) :
(μ.restrict s) t = 0 μ (t s) = 0
theorem measure_theory.measure.measure_inter_eq_zero_of_restrict {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (h : (μ.restrict s) t = 0) :
μ (t s) = 0
theorem measure_theory.measure.restrict_apply_eq_zero' {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) :
(μ.restrict s) t = 0 μ (t s) = 0
@[simp]
theorem measure_theory.measure.restrict_eq_zero {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} :
μ.restrict s = 0 μ s = 0
theorem measure_theory.measure.restrict_zero_set {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (h : μ s = 0) :
μ.restrict s = 0
@[simp]
theorem measure_theory.measure.restrict_eq_self_of_measurable_subset {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (ht : measurable_set t) (t_subset : t s) :
(μ.restrict s) t = μ t
theorem measure_theory.measure.restrict_union_apply {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s s' t : set α} (h : disjoint (t s) (t s')) (hs : measurable_set s) (hs' : measurable_set s') (ht : measurable_set t) :
(μ.restrict (s s')) t = (μ.restrict s) t + (μ.restrict s') t
theorem measure_theory.measure.restrict_union {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (h : disjoint s t) (hs : measurable_set s) (ht : measurable_set t) :
μ.restrict (s t) = μ.restrict s + μ.restrict t
theorem measure_theory.measure.restrict_union_add_inter {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (ht : measurable_set t) :
μ.restrict (s t) + μ.restrict (s t) = μ.restrict s + μ.restrict t
@[simp]
@[simp]
theorem measure_theory.measure.restrict_union_le {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} (s s' : set α) :
μ.restrict (s s') μ.restrict s + μ.restrict s'
theorem measure_theory.measure.restrict_Union_apply {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} [encodable ι] {s : ι → set α} (hd : pairwise (disjoint on s)) (hm : ∀ (i : ι), measurable_set (s i)) {t : set α} (ht : measurable_set t) :
(μ.restrict (⋃ (i : ι), s i)) t = ∑' (i : ι), (μ.restrict (s i)) t
theorem measure_theory.measure.restrict_Union_apply_eq_supr {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} [encodable ι] {s : ι → set α} (hm : ∀ (i : ι), measurable_set (s i)) (hd : directed has_subset.subset s) {t : set α} (ht : measurable_set t) :
(μ.restrict (⋃ (i : ι), s i)) t = ⨆ (i : ι), (μ.restrict (s i)) t
theorem measure_theory.measure.restrict_map {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : measurable f) {s : set β} (hs : measurable_set s) :
theorem measure_theory.measure.restrict_mono' {α : Type u_1} [measurable_space α] ⦃s s' : set α⦄ ⦃μ ν : measure_theory.measure α⦄ (hs : s ≤ᵐ[μ] s') (hμν : μ ν) :
μ.restrict s ν.restrict s'

Restriction of a measure to a subset is monotone both in set and in measure.

theorem measure_theory.measure.restrict_mono {α : Type u_1} [measurable_space α] ⦃s s' : set α⦄ (hs : s s') ⦃μ ν : measure_theory.measure α⦄ (hμν : μ ν) :
μ.restrict s ν.restrict s'

Restriction of a measure to a subset is monotone both in set and in measure.

theorem measure_theory.measure.restrict_le_self {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} :
μ.restrict s μ
theorem measure_theory.measure.restrict_congr_meas {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {s : set α} (hs : measurable_set s) :
μ.restrict s = ν.restrict s ∀ (t : set α), t smeasurable_set tμ t = ν t
theorem measure_theory.measure.restrict_congr_mono {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {s t : set α} (hs : s t) (hm : measurable_set s) (h : μ.restrict t = ν.restrict t) :
μ.restrict s = ν.restrict s
theorem measure_theory.measure.restrict_union_congr {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {s t : set α} (hsm : measurable_set s) (htm : measurable_set t) :
μ.restrict (s t) = ν.restrict (s t) μ.restrict s = ν.restrict s μ.restrict t = ν.restrict t

If two measures agree on all measurable subsets of s and t, then they agree on all measurable subsets of s ∪ t.

theorem measure_theory.measure.restrict_finset_bUnion_congr {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ ν : measure_theory.measure α} {s : finset ι} {t : ι → set α} (htm : ∀ (i : ι), i smeasurable_set (t i)) :
μ.restrict (⋃ (i : ι) (H : i s), t i) = ν.restrict (⋃ (i : ι) (H : i s), t i) ∀ (i : ι), i sμ.restrict (t i) = ν.restrict (t i)
theorem measure_theory.measure.restrict_Union_congr {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ ν : measure_theory.measure α} [encodable ι] {s : ι → set α} (hm : ∀ (i : ι), measurable_set (s i)) :
μ.restrict (⋃ (i : ι), s i) = ν.restrict (⋃ (i : ι), s i) ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)
theorem measure_theory.measure.restrict_bUnion_congr {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ ν : measure_theory.measure α} {s : set ι} {t : ι → set α} (hc : s.countable) (htm : ∀ (i : ι), i smeasurable_set (t i)) :
μ.restrict (⋃ (i : ι) (H : i s), t i) = ν.restrict (⋃ (i : ι) (H : i s), t i) ∀ (i : ι), i sμ.restrict (t i) = ν.restrict (t i)
theorem measure_theory.measure.restrict_sUnion_congr {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {S : set (set α)} (hc : S.countable) (hm : ∀ (s : set α), s Smeasurable_set s) :
μ.restrict (⋃₀S) = ν.restrict (⋃₀S) ∀ (s : set α), s Sμ.restrict s = ν.restrict s

This lemma shows that restrict and to_outer_measure commute. Note that the LHS has a restrict on measures and the RHS has a restrict on outer measures.

theorem measure_theory.measure.restrict_Inf_eq_Inf_restrict {α : Type u_1} [measurable_space α] {t : set α} {m : set (measure_theory.measure α)} (hm : m.nonempty) (ht : measurable_set t) :
(Inf m).restrict t = Inf ((λ (μ : measure_theory.measure α), μ.restrict t) '' m)

This lemma shows that Inf and restrict commute for measures.

theorem measure_theory.measure.restrict_apply' {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) :
(μ.restrict s) t = μ (t s)

If s is a measurable set, then the outer measure of t with respect to the restriction of the measure to s equals the outer measure of t ∩ s. This is an alternate version of measure.restrict_apply, requiring that s is measurable instead of t.

theorem measure_theory.measure.restrict_eq_self_of_subset_of_measurable {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (t_subset : t s) :
(μ.restrict s) t = μ t

Extensionality results #

theorem measure_theory.measure.ext_iff_of_Union_eq_univ {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ ν : measure_theory.measure α} [encodable ι] {s : ι → set α} (hm : ∀ (i : ι), measurable_set (s i)) (hs : (⋃ (i : ι), s i) = set.univ) :
μ = ν ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)

Two measures are equal if they have equal restrictions on a spanning collection of sets (formulated using Union).

theorem measure_theory.measure.ext_of_Union_eq_univ {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ ν : measure_theory.measure α} [encodable ι] {s : ι → set α} (hm : ∀ (i : ι), measurable_set (s i)) (hs : (⋃ (i : ι), s i) = set.univ) :
(∀ (i : ι), μ.restrict (s i) = ν.restrict (s i))μ = ν

Alias of ext_iff_of_Union_eq_univ.

theorem measure_theory.measure.ext_iff_of_bUnion_eq_univ {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ ν : measure_theory.measure α} {S : set ι} {s : ι → set α} (hc : S.countable) (hm : ∀ (i : ι), i Smeasurable_set (s i)) (hs : (⋃ (i : ι) (H : i S), s i) = set.univ) :
μ = ν ∀ (i : ι), i Sμ.restrict (s i) = ν.restrict (s i)

Two measures are equal if they have equal restrictions on a spanning collection of sets (formulated using bUnion).

theorem measure_theory.measure.ext_of_bUnion_eq_univ {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ ν : measure_theory.measure α} {S : set ι} {s : ι → set α} (hc : S.countable) (hm : ∀ (i : ι), i Smeasurable_set (s i)) (hs : (⋃ (i : ι) (H : i S), s i) = set.univ) :
(∀ (i : ι), i Sμ.restrict (s i) = ν.restrict (s i))μ = ν

Alias of ext_iff_of_bUnion_eq_univ.

theorem measure_theory.measure.ext_iff_of_sUnion_eq_univ {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {S : set (set α)} (hc : S.countable) (hm : ∀ (s : set α), s Smeasurable_set s) (hs : ⋃₀S = set.univ) :
μ = ν ∀ (s : set α), s Sμ.restrict s = ν.restrict s

Two measures are equal if they have equal restrictions on a spanning collection of sets (formulated using sUnion).

theorem measure_theory.measure.ext_of_sUnion_eq_univ {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {S : set (set α)} (hc : S.countable) (hm : ∀ (s : set α), s Smeasurable_set s) (hs : ⋃₀S = set.univ) :
(∀ (s : set α), s Sμ.restrict s = ν.restrict s)μ = ν

Alias of ext_iff_of_sUnion_eq_univ.

theorem measure_theory.measure.ext_of_generate_from_of_cover {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {S T : set (set α)} (h_gen : _inst_1 = measurable_space.generate_from S) (hc : T.countable) (h_inter : is_pi_system S) (hm : ∀ (t : set α), t Tmeasurable_set t) (hU : ⋃₀T = 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 measure_theory.measure.ext_of_generate_from_of_cover_subset {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {S T : set (set α)} (h_gen : _inst_1 = measurable_space.generate_from S) (h_inter : is_pi_system S) (h_sub : T S) (hc : T.countable) (hU : ⋃₀T = 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 a increasing spanning sequence of sets in the π-system. This lemma is formulated using sUnion.

theorem measure_theory.measure.ext_of_generate_from_of_Union {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} (C : set (set α)) (B : set α) (hA : _inst_1 = measurable_space.generate_from C) (hC : is_pi_system C) (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 a increasing spanning sequence of sets in the π-system. This lemma is formulated using Union. finite_spanning_sets_in.ext is a reformulation of this lemma.

theorem measure_theory.measure.le_dirac_apply {α : Type u_1} [measurable_space α] {s : set α} {a : α} :
@[simp]
theorem measure_theory.measure.dirac_apply' {α : Type u_1} [measurable_space α] {s : set α} (a : α) (hs : measurable_set s) :
@[simp]
theorem measure_theory.measure.dirac_apply_of_mem {α : Type u_1} [measurable_space α] {s : set α} {a : α} (h : a s) :
@[simp]
theorem measure_theory.measure.map_dirac {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {f : α → β} (hf : measurable f) (a : α) :
def measure_theory.measure.sum {α : Type u_1} {ι : Type u_5} [measurable_space α] (f : ι → measure_theory.measure α) :

Sum of an indexed family of measures.

Equations
theorem measure_theory.measure.le_sum_apply {α : Type u_1} {ι : Type u_5} [measurable_space α] (f : ι → measure_theory.measure α) (s : set α) :
∑' (i : ι), (f i) s (measure_theory.measure.sum f) s
@[simp]
theorem measure_theory.measure.sum_apply {α : Type u_1} {ι : Type u_5} [measurable_space α] (f : ι → measure_theory.measure α) {s : set α} (hs : measurable_set s) :
(measure_theory.measure.sum f) s = ∑' (i : ι), (f i) s
theorem measure_theory.measure.le_sum {α : Type u_1} {ι : Type u_5} [measurable_space α] (μ : ι → measure_theory.measure α) (i : ι) :
theorem measure_theory.measure.restrict_Union {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} [encodable ι] {s : ι → set α} (hd : pairwise (disjoint on s)) (hm : ∀ (i : ι), measurable_set (s i)) :
μ.restrict (⋃ (i : ι), s i) = measure_theory.measure.sum (λ (i : ι), μ.restrict (s i))
theorem measure_theory.measure.restrict_Union_le {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} [encodable ι] {s : ι → set α} :
μ.restrict (⋃ (i : ι), s i) measure_theory.measure.sum (λ (i : ι), μ.restrict (s i))
@[simp]
theorem measure_theory.measure.sum_cond {α : Type u_1} [measurable_space α] (μ ν : measure_theory.measure α) :
measure_theory.measure.sum (λ (b : bool), cond b μ ν) = μ + ν
@[simp]
theorem measure_theory.measure.restrict_sum {α : Type u_1} {ι : Type u_5} [measurable_space α] (μ : ι → measure_theory.measure α) {s : set α} (hs : measurable_set s) :
theorem measure_theory.measure.le_count_apply {α : Type u_1} [measurable_space α] {s : set α} :
theorem measure_theory.measure.count_apply {α : Type u_1} [measurable_space α] {s : set α} (hs : measurable_set s) :

count measure evaluates to infinity at infinite sets.

Absolute continuity #

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.

Equations
theorem measure_theory.measure.absolutely_continuous_of_le {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} (h : μ ν) :
μ ν
theorem has_le.le.absolutely_continuous {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} (h : μ ν) :
μ ν

Alias of absolutely_continuous_of_le.

theorem measure_theory.measure.absolutely_continuous_of_eq {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} (h : μ = ν) :
μ ν
theorem eq.absolutely_continuous {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} (h : μ = ν) :
μ ν

Alias of absolutely_continuous_of_eq.

theorem measure_theory.measure.absolutely_continuous.mk {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} (h : ∀ ⦃s : set α⦄, measurable_set sν s = 0μ s = 0) :
μ ν
theorem measure_theory.measure.absolutely_continuous.trans {α : Type u_1} [measurable_space α] {μ₁ μ₂ μ₃ : measure_theory.measure α} (h1 : μ₁ μ₂) (h2 : μ₂ μ₃) :
μ₁ μ₃
theorem measure_theory.measure.absolutely_continuous.map {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ ν : measure_theory.measure α} (h : μ ν) (f : α → β) :
theorem has_le.le.absolutely_continuous_of_ae {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} :
μ.ae ν.aeμ ν

Alias of ae_le_iff_absolutely_continuous.

theorem measure_theory.measure.absolutely_continuous.ae_le {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} :
μ νμ.ae ν.ae

Alias of ae_le_iff_absolutely_continuous.

theorem ae_mono' {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} :
μ νμ.ae ν.ae

Alias of absolutely_continuous.ae_le.

theorem measure_theory.measure.absolutely_continuous.ae_eq {α : Type u_1} {δ : Type u_4} [measurable_space α] {μ ν : measure_theory.measure α} (h : μ ν) {f g : α → δ} (h' : f =ᵐ[ν] g) :
f =ᵐ[μ] g

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

structure measure_theory.measure.quasi_measure_preserving {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] (f : α → β) (μa : measure_theory.measure α . "volume_tac") (μb : measure_theory.measure β . "volume_tac") :
Prop

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.

theorem measure_theory.measure.quasi_measure_preserving.mono {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μa μa' : measure_theory.measure α} {μb μb' : measure_theory.measure β} {f : α → β} (ha : μa' μa) (hb : μb μb') (h : measure_theory.measure.quasi_measure_preserving f μa μb) :
theorem measure_theory.measure.quasi_measure_preserving.ae {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μa : measure_theory.measure α} {μb : measure_theory.measure β} {f : α → β} (h : measure_theory.measure.quasi_measure_preserving f μa μb) {p : β → Prop} (hg : ∀ᵐ (x : β) ∂μb, p x) :
∀ᵐ (x : α) ∂μa, p (f x)
theorem measure_theory.measure.quasi_measure_preserving.ae_eq {α : Type u_1} {β : Type u_2} {δ : Type u_4} [measurable_space α] [measurable_space β] {μa : measure_theory.measure α} {μb : measure_theory.measure β} {f : α → β} (h : measure_theory.measure.quasi_measure_preserving f μa μb) {g₁ g₂ : β → δ} (hg : g₁ =ᵐ[μb] g₂) :
g₁ f =ᵐ[μa] g₂ f

The cofinite filter #

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

Equations
theorem measure_theory.measure.mem_cofinite {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} :
theorem measure_theory.measure.eventually_cofinite {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {p : α → Prop} :
(∀ᶠ (x : α) in μ.cofinite, p x) μ {x : α | ¬p x} <
@[simp]
theorem measure_theory.ae_eq_bot {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} :
μ.ae = μ = 0
@[simp]
theorem measure_theory.ae_ne_bot {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} :
μ.ae.ne_bot μ 0
@[simp]
theorem measure_theory.ae_zero {α : Type u_1} [measurable_space α] :
0.ae =
theorem measure_theory.ae_mono {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} (h : μ ν) :
μ.ae ν.ae
theorem measure_theory.mem_ae_map_iff {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : measurable f) {s : set β} (hs : measurable_set s) :
theorem measure_theory.mem_ae_of_mem_ae_map {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : measurable f) {s : set β} (hs : s ((measure_theory.measure.map f) μ).ae) :
f ⁻¹' s μ.ae
theorem measure_theory.ae_map_iff {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : measurable f) {p : β → Prop} (hp : measurable_set {x : β | p x}) :
(∀ᵐ (y : β) ∂(measure_theory.measure.map f) μ, p y) ∀ᵐ (x : α) ∂μ, p (f x)
theorem measure_theory.ae_of_ae_map {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} (hf : measurable f) {p : β → Prop} (h : ∀ᵐ (y : β) ∂(measure_theory.measure.map f) μ, p y) :
∀ᵐ (x : α) ∂μ, p (f x)
theorem measure_theory.ae_map_mem_range {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] (f : α → β) (hf : measurable_set (set.range f)) (μ : measure_theory.measure α) :
∀ᵐ (x : β) ∂(measure_theory.measure.map f) μ, x set.range f
theorem measure_theory.ae_restrict_iff {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {p : α → Prop} (hp : measurable_set {x : α | p x}) :
(∀ᵐ (x : α) ∂μ.restrict s, p x) ∀ᵐ (x : α) ∂μ, x sp x
theorem measure_theory.ae_imp_of_ae_restrict {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {p : α → Prop} (h : ∀ᵐ (x : α) ∂μ.restrict s, p x) :
∀ᵐ (x : α) ∂μ, x sp x
theorem measure_theory.ae_restrict_iff' {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {p : α → Prop} (hs : measurable_set s) :
(∀ᵐ (x : α) ∂μ.restrict s, p x) ∀ᵐ (x : α) ∂μ, x sp x
theorem measure_theory.ae_restrict_mem {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) :
∀ᵐ (x : α) ∂μ.restrict s, x s
theorem measure_theory.ae_restrict_of_ae {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {p : α → Prop} (h : ∀ᵐ (x : α) ∂μ, p x) :
∀ᵐ (x : α) ∂μ.restrict s, p x
theorem measure_theory.ae_restrict_of_ae_restrict_of_subset {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} {p : α → Prop} (hst : s t) (h : ∀ᵐ (x : α) ∂μ.restrict t, p x) :
∀ᵐ (x : α) ∂μ.restrict s, p x
theorem measure_theory.ae_smul_measure {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {p : α → Prop} (h : ∀ᵐ (x : α) ∂μ, p x) (c : ℝ≥0∞) :
∀ᵐ (x : α) ∂c μ, p x
theorem measure_theory.ae_smul_measure_iff {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {p : α → Prop} {c : ℝ≥0∞} (hc : c 0) :
(∀ᵐ (x : α) ∂c μ, p x) ∀ᵐ (x : α) ∂μ, p x
theorem measure_theory.ae_add_measure_iff {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {p : α → Prop} {ν : measure_theory.measure α} :
(∀ᵐ (x : α) ∂μ + ν, p x) (∀ᵐ (x : α) ∂μ, p x) ∀ᵐ (x : α) ∂ν, p x
theorem measure_theory.ae_eq_comp' {α : Type u_1} {β : Type u_2} {δ : Type u_4} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} {f : α → β} {g g' : β → δ} (hf : measurable f) (h : g =ᵐ[ν] g') (h2 : (measure_theory.measure.map f) μ ν) :
g f =ᵐ[μ] g' f
theorem measure_theory.ae_eq_comp {α : Type u_1} {β : Type u_2} {δ : Type u_4} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} {g g' : β → δ} (hf : measurable f) (h : g =ᵐ[(measure_theory.measure.map f) μ] g') :
g f =ᵐ[μ] g' f
theorem measure_theory.le_ae_restrict {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} :
μ.ae 𝓟 s (μ.restrict s).ae
@[simp]
theorem measure_theory.ae_restrict_eq {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) :
(μ.restrict s).ae = μ.ae 𝓟 s
@[simp]
theorem measure_theory.ae_restrict_eq_bot {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} :
(μ.restrict s).ae = μ s = 0
@[simp]
theorem measure_theory.ae_restrict_ne_bot {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} :
(μ.restrict s).ae.ne_bot 0 < μ s
theorem measure_theory.self_mem_ae_restrict {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) :
s (μ.restrict s).ae
theorem measure_theory.ae_eventually_not_mem {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : ∀ (i : ), measurable_set (s i)) (hs' : ∑' (i : ), μ (s i) ) :
∀ᵐ (x : α) ∂μ, ∀ᶠ (n : ) in filter.at_top, x s n

A version of the Borel-Cantelli lemma: if sᵢ is a sequence of measurable sets such that ∑ μ sᵢ exists, then for almost all x, x does not belong to almost all sᵢ.

theorem measure_theory.mem_ae_dirac_iff {α : Type u_1} [measurable_space α] {s : set α} {a : α} (hs : measurable_set s) :
theorem measure_theory.ae_dirac_iff {α : Type u_1} [measurable_space α] {a : α} {p : α → Prop} (hp : measurable_set {x : α | p x}) :
(∀ᵐ (x : α) ∂measure_theory.measure.dirac a, p x) p a
theorem measure_theory.ae_eq_dirac' {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] [measurable_singleton_class β] {a : α} {f : α → β} (hf : measurable f) :
theorem measure_theory.ae_eq_dirac {α : Type u_1} {δ : Type u_4} [measurable_space α] [measurable_singleton_class α] {a : α} (f : α → δ) :
theorem measure_theory.restrict_mono_ae {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (h : s ≤ᵐ[μ] t) :
theorem measure_theory.restrict_congr_set {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (H : s =ᵐ[μ] t) :
μ.restrict s = μ.restrict t
@[instance]

The measure of the whole space with respect to a finite measure, considered as ℝ≥0.

Equations
theorem measure_theory.measure.le_of_add_le_add_left {α : Type u_1} [measurable_space α] {μ ν₁ ν₂ : measure_theory.measure α} [measure_theory.finite_measure μ] (A2 : μ + ν₁ μ + ν₂) :
ν₁ ν₂

le_of_add_le_add_left is normally applicable to ordered_cancel_add_comm_monoid, but it holds for measures with the additional assumption that μ is finite.

theorem measure_theory.summable_measure_to_real {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} [hμ : measure_theory.finite_measure μ] {f : set α} (hf₁ : ∀ (i : ), measurable_set (f i)) (hf₂ : pairwise (disjoint on f)) :
summable (λ (x : ), (μ (f x)).to_real)
@[class]
structure measure_theory.probability_measure {α : Type u_1} [measurable_space α] (μ : measure_theory.measure α) :
Prop

A measure μ is called a probability measure if μ univ = 1.

Instances
@[class]
structure measure_theory.has_no_atoms {α : Type u_1} [measurable_space α] (μ : measure_theory.measure α) :
Prop
  • measure_singleton : ∀ (x : α), μ {x} = 0

Measure μ has no atoms if the measure of each singleton is zero.

NB: Wikipedia assumes that for any measurable set s with positive μ-measure, there exists a measurable t ⊆ s such that 0 < μ t < μ s. While this implies μ {x} = 0, the converse is not true.

Instances
theorem set.subsingleton.measure_eq {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} [measure_theory.has_no_atoms μ] (hs : s.subsingleton) :
μ s = 0

Alias of measure_subsingleton.

theorem set.countable.measure_zero {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} [measure_theory.has_no_atoms μ] (h : s.countable) :
μ s = 0
theorem set.finite.measure_zero {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} [measure_theory.has_no_atoms μ] (h : s.finite) :
μ s = 0
theorem finset.measure_zero {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} [measure_theory.has_no_atoms μ] (s : finset α) :
μ s = 0
theorem measure_theory.insert_ae_eq_self {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} [measure_theory.has_no_atoms μ] (a : α) (s : set α) :
insert a s =ᵐ[μ] s
theorem measure_theory.ite_ae_eq_of_measure_zero {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {γ : Type u_2} (f g : α → γ) (s : set α) (hs_zero : μ s = 0) :
(λ (x : α), ite (x s) (f x) (g x)) =ᵐ[μ] g
theorem measure_theory.ite_ae_eq_of_measure_compl_zero {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {γ : Type u_2} (f g : α → γ) (s : set α) (hs_zero : μ s = 0) :
(λ (x : α), ite (x s) (f x) (g x)) =ᵐ[μ] f
def measure_theory.measure.finite_at_filter {α : Type u_1} [measurable_space α] (μ : measure_theory.measure α) (f : filter α) :
Prop

A measure is called finite at filter f if it is finite at some set s ∈ f. Equivalently, it is eventually finite at s in f.lift' powerset.

Equations
theorem measure_theory.measure.finite_at_filter.exists_mem_basis {α : Type u_1} {ι : Type u_5} [measurable_space α] {μ : measure_theory.measure α} {f : filter α} (hμ : μ.finite_at_filter f) {p : ι → Prop} {s : ι → set α} (hf : f.has_basis p s) :
∃ (i : ι) (hi : p i), μ (s i) <
@[nolint]
structure measure_theory.measure.finite_spanning_sets_in {α : Type u_1} [measurable_space α] (μ : measure_theory.measure α) (C : set (set α)) :
Type u_1

μ has finite spanning sets in C if there is a countable sequence of sets in C that have finite measures. This structure is a type, which is useful if we want to record extra properties about the sets, such as that they are monotone. sigma_finite is defined in terms of this: μ is σ-finite if there exists a sequence of finite spanning sets in the collection of all measurable sets.

If μ is σ-finite it has finite spanning sets in the collection of all measurable sets.

Equations

A noncomputable way to get a monotone collection of sets that span univ and have finite measure using classical.some. This definition satisfies monotonicity in addition to all other properties in sigma_finite.

Equations

If μ has finite spanning sets in C and C ⊆ D then μ has finite spanning sets in D.

Equations

If μ has finite spanning sets in the collection of measurable sets C, then μ is σ-finite.

theorem measure_theory.measure.finite_spanning_sets_in.ext {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {C : set (set α)} (hA : _inst_1 = measurable_space.generate_from C) (hC : is_pi_system C) (h : μ.finite_spanning_sets_in C) (h_eq : ∀ (s : set α), s Cμ s = ν s) :
μ = ν

An extensionality for measures. It is ext_of_generate_from_of_Union formulated in terms of finite_spanning_sets_in.

theorem measure_theory.measure.sigma_finite_of_countable {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {S : set (set α)} (hc : S.countable) (hμ : ∀ (s : set α), s Sμ s < ) (hU : ⋃₀S = set.univ) :
@[instance]

Every finite measure is σ-finite.

@[instance]
@[class]

A measure is called locally finite if it is finite in some neighborhood of each point.

Instances
theorem measure_theory.ext_on_measurable_space_of_generate_finite {α : Type u_1} (m₀ : measurable_space α) {μ ν : measure_theory.measure α} [measure_theory.finite_measure μ] (C : set (set α)) (hμν : ∀ (s : set α), s Cμ s = ν s) {m : measurable_space α} (h : m m₀) (hA : m = measurable_space.generate_from C) (hC : is_pi_system C) (h_univ : μ set.univ = ν set.univ) {s : set α} (hs : m.measurable_set' s) :
μ s = ν s

If two finite measures give the same mass to the whole space and coincide on a π-system made of measurable sets, then they coincide on all sets in the σ-algebra generated by the π-system.

theorem measure_theory.ext_of_generate_finite {α : Type u_1} [measurable_space α] (C : set (set α)) (hA : _inst_1 = measurable_space.generate_from C) (hC : is_pi_system C) {μ ν : measure_theory.measure α} [measure_theory.finite_measure μ] (hμν : ∀ (s : set α), s Cμ s = ν s) (h_univ : μ set.univ = ν set.univ) :
μ = ν

Two finite measures are equal if they are equal on the π-system generating the σ-algebra (and univ).

Alias of inf_ae_iff.

theorem measure_theory.measure.finite_at_filter.mono {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {f g : filter α} (hf : f g) (hμ : μ ν) :
theorem measure_theory.measure.finite_at_filter.eventually {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : filter α} (h : μ.finite_at_filter f) :
∀ᶠ (s : set α) in f.lift' set.powerset, μ s <
@[simp]

Subtraction of measures #

@[instance]

The measure μ - ν is defined to be the least measure τ such that μ ≤ τ + ν. It is the equivalent of (μ - ν) ⊔ 0 if μ and ν were signed measures. Compare with ennreal.has_sub. Specifically, note that if you have α = {1,2}, and μ {1} = 2, μ {2} = 0, and ν {2} = 2, ν {1} = 0, then (μ - ν) {1, 2} = 2. However, if μ ≤ ν, and ν univ ≠ ∞, then (μ - ν) + ν = μ.

Equations
theorem measure_theory.measure.sub_def {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} :
μ - ν = Inf {d : measure_theory.measure α | μ d + ν}
theorem measure_theory.measure.sub_eq_zero_of_le {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} (h : μ ν) :
μ - ν = 0
theorem measure_theory.measure.sub_apply {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {s : set α} [measure_theory.finite_measure ν] (h₁ : measurable_set s) (h₂ : ν μ) :
- ν) s = μ s - ν s

This application lemma only works in special circumstances. Given knowledge of when μ ≤ ν and ν ≤ μ, a more general application lemma can be written.

theorem measure_theory.measure.sub_add_cancel_of_le {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} [measure_theory.finite_measure ν] (h₁ : ν μ) :
μ - ν + ν = μ
theorem measure_theory.measure.sub_le {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} :
μ - ν μ
theorem measure_theory.measure.restrict_sub_eq_restrict_sub_restrict {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {s : set α} (h_meas_s : measurable_set s) :
- ν).restrict s = μ.restrict s - ν.restrict s
theorem measure_theory.measure.sub_apply_eq_zero_of_restrict_le_restrict {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} {s : set α} (h_le : μ.restrict s ν.restrict s) (h_meas_s : measurable_set s) :
- ν) s = 0

Interactions of measurable equivalences and measures

theorem measurable_equiv.map_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} (f : α ≃ᵐ β) (s : set β) :

If we map a measure along a measurable equivalence, we can compute the measure on all sets (not just the measurable ones).

@[simp]
@[simp]
@[class]
structure measure_theory.measure.is_complete {α : Type u_1} {_x : measurable_space α} (μ : measure_theory.measure α) :
Prop

A measure is complete if every null set is also measurable. A null set is a subset of a measurable set with measure 0. Since every measure is defined as a special case of an outer measure, we can more simply state that a set s is null if μ s = 0.

Instances
theorem measure_theory.measure.is_complete_iff {α : Type u_1} {_x : measurable_space α} {μ : measure_theory.measure α} :
μ.is_complete ∀ (s : set α), μ s = 0measurable_set s
theorem measure_theory.measure.is_complete.out {α : Type u_1} {_x : measurable_space α} {μ : measure_theory.measure α} (h : μ.is_complete) (s : set α) :
μ s = 0measurable_set s
def null_measurable_set {α : Type u_1} [measurable_space α] (μ : measure_theory.measure α) (s : set α) :
Prop

A set is null measurable if it is the union of a null set and a measurable set.

Equations
theorem null_measurable_set_iff {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} :
null_measurable_set μ s ∃ (t : set α), t s measurable_set t μ (s \ t) = 0
theorem null_measurable_set_measure_eq {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (st : t s) (hz : μ (s \ t) = 0) :
μ s = μ t
theorem null_measurable_set.union_null {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s z : set α} (hs : null_measurable_set μ s) (hz : μ z = 0) :
theorem null_null_measurable_set {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {z : set α} (hz : μ z = 0) :
theorem null_measurable_set.Union_nat {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : ∀ (i : ), null_measurable_set μ (s i)) :
theorem measurable_set.diff_null {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s z : set α} (hs : measurable_set s) (hz : μ z = 0) :
theorem null_measurable_set.diff_null {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s z : set α} (hs : null_measurable_set μ s) (hz : μ z = 0) :
theorem null_measurable_set.compl {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : null_measurable_set μ s) :
theorem null_measurable_set_iff_ae {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} :
null_measurable_set μ s ∃ (t : set α), measurable_set t s =ᵐ[μ] t
theorem null_measurable_set_iff_sandwich {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} :
null_measurable_set μ s ∃ (t u : set α), measurable_set t measurable_set u t s s u μ (u \ t) = 0
theorem restrict_apply_of_null_measurable_set {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} (ht : null_measurable_set (μ.restrict s) t) :
(μ.restrict s) t = μ (t s)
def null_measurable {α : Type u_1} [measurable_space α] (μ : measure_theory.measure α) :

The measurable space of all null measurable sets.

Equations

Given a measure we can complete it to a (complete) measure on all null measurable sets.

Equations
@[instance]
theorem measurable.ae_eq {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} [hμ : μ.is_complete] {f g : α → β} (hf : measurable f) (hfg : f =ᵐ[μ] g) :
def measure_theory.measure.trim {α : Type u_1} {m m0 : measurable_space α} (μ : measure_theory.measure α) (hm : m