mathlib3 documentation

measure_theory.measure.measure_space

Measure spaces #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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.ae_uIoc_iff {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} [linear_order 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 measure_theory.ae_restrict_uIoc_iff.

theorem measure_theory.measure_union {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s₁ s₂ : set α} (hd : disjoint s₁ s₂) (h : measurable_set s₂) :
μ (s₁ s₂) = μ s₁ + μ s₂
theorem measure_theory.measure_union' {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s₁ s₂ : set α} (hd : disjoint s₁ s₂) (h : measurable_set s₁) :
μ (s₁ s₂) = μ s₁ + μ s₂
theorem measure_theory.measure_inter_add_diff {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {t : set α} (s : set α) (ht : measurable_set t) :
μ (s t) + μ (s \ t) = μ s
theorem measure_theory.measure_diff_add_inter {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {t : set α} (s : set α) (ht : measurable_set t) :
μ (s \ t) + μ (s t) = μ s
theorem measure_theory.measure_union_add_inter {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {t : set α} (s : set α) (ht : measurable_set t) :
μ (s t) + μ (s t) = μ s + μ t
theorem measure_theory.measure_union_add_inter' {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) (t : set α) :
μ (s t) + μ (s t) = μ s + μ t
theorem measure_theory.measure_bUnion₀ {α : Type u_1} {β : Type u_2} {m : measurable_space α} {μ : measure_theory.measure α} {s : set β} {f : β set α} (hs : s.countable) (hd : s.pairwise (measure_theory.ae_disjoint μ on f)) (h : (b : β), b s measure_theory.null_measurable_set (f b) μ) :
μ ( (b : β) (H : b s), f b) = ∑' (p : s), μ (f p)
theorem measure_theory.measure_bUnion {α : Type u_1} {β : Type u_2} {m : measurable_space α} {μ : measure_theory.measure α} {s : set β} {f : β set α} (hs : s.countable) (hd : s.pairwise_disjoint f) (h : (b : β), b s measurable_set (f b)) :
μ ( (b : β) (H : b s), f b) = ∑' (p : s), μ (f p)
theorem measure_theory.measure_sUnion₀ {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {S : set (set α)} (hs : S.countable) (hd : S.pairwise (measure_theory.ae_disjoint μ)) (h : (s : set α), s S measure_theory.null_measurable_set s μ) :
μ (⋃₀ S) = ∑' (s : S), μ s
theorem measure_theory.measure_sUnion {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {S : set (set α)} (hs : S.countable) (hd : S.pairwise disjoint) (h : (s : set α), s S measurable_set s) :
μ (⋃₀ S) = ∑' (s : S), μ s
theorem measure_theory.measure_bUnion_finset₀ {α : Type u_1} {ι : Type u_5} {m : measurable_space α} {μ : measure_theory.measure α} {s : finset ι} {f : ι set α} (hd : s.pairwise (measure_theory.ae_disjoint μ on f)) (hm : (b : ι), b s measure_theory.null_measurable_set (f b) μ) :
μ ( (b : ι) (H : b s), f b) = s.sum (λ (p : ι), μ (f p))
theorem measure_theory.measure_bUnion_finset {α : Type u_1} {ι : Type u_5} {m : measurable_space α} {μ : measure_theory.measure α} {s : finset ι} {f : ι set α} (hd : s.pairwise_disjoint f) (hm : (b : ι), b s measurable_set (f b)) :
μ ( (b : ι) (H : b s), f b) = s.sum (λ (p : ι), μ (f p))
theorem measure_theory.tsum_meas_le_meas_Union_of_disjoint {α : Type u_1} {ι : Type u_2} [measurable_space α] (μ : measure_theory.measure α) {As : ι set α} (As_mble : (i : ι), measurable_set (As i)) (As_disj : pairwise (disjoint on As)) :
∑' (i : ι), μ (As i) μ ( (i : ι), As i)

The measure of a disjoint union (even uncountable) of measurable sets is at least the sum of the measures of the sets.

theorem measure_theory.tsum_measure_preimage_singleton {α : Type u_1} {β : Type u_2} {m : measurable_space α} {μ : measure_theory.measure α} {s : set β} (hs : s.countable) {f : α β} (hf : (y : β), y s measurable_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} {m : measurable_space α} {μ : measure_theory.measure α} (s : finset β) {f : α β} (hf : (y : β), y s measurable_set (f ⁻¹' {y})) :
s.sum (λ (b : β), 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} {m : 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} {m : 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_add_diff {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) (t : set α) :
μ s + μ (t \ s) = μ (s t)
theorem measure_theory.measure_diff' {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {t : set α} (s : set α) (hm : measurable_set t) (h_fin : μ t ) :
μ (s \ t) = μ (s t) - μ t
theorem measure_theory.measure_diff {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s₁ s₂ : set α} (h : s₂ s₁) (h₂ : measurable_set s₂) (h_fin : μ s₂ ) :
μ (s₁ \ s₂) = μ s₁ - μ s₂
theorem measure_theory.le_measure_diff {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s₁ s₂ : set α} :
μ s₁ - μ s₂ μ (s₁ \ s₂)
theorem measure_theory.measure_diff_lt_of_lt_add {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (hst : s t) (hs' : μ s ) {ε : ennreal} (h : μ t < μ s + ε) :
μ (t \ s) < ε
theorem measure_theory.measure_diff_le_iff_le_add {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (hst : s t) (hs' : μ s ) {ε : ennreal} :
μ (t \ s) ε μ t μ s + ε
theorem measure_theory.measure_eq_measure_of_null_diff {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (hst : s t) (h_nulldiff : μ (t \ s) = 0) :
μ s = μ t
theorem measure_theory.measure_eq_measure_of_between_null_diff {α : Type u_1} {m : 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.measure_eq_measure_smaller_of_between_null_diff {α : Type u_1} {m : 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
theorem measure_theory.measure_eq_measure_larger_of_between_null_diff {α : Type u_1} {m : 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 s₂ = μ s₃
theorem measure_theory.measure_compl {α : Type u_1} {m : 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
@[simp]
theorem measure_theory.union_ae_eq_left_iff_ae_subset {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s t : set α} :
s t =ᵐ[μ] s t ≤ᵐ[μ] s
@[simp]
theorem measure_theory.union_ae_eq_right_iff_ae_subset {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s t : set α} :
s t =ᵐ[μ] t s ≤ᵐ[μ] t
theorem measure_theory.ae_eq_of_ae_subset_of_measure_ge {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (h₁ : s ≤ᵐ[μ] t) (h₂ : μ t μ s) (hsm : measurable_set s) (ht : μ t ) :
s =ᵐ[μ] t
theorem measure_theory.ae_eq_of_subset_of_measure_ge {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (h₁ : s t) (h₂ : μ t μ s) (hsm : measurable_set s) (ht : μ t ) :
s =ᵐ[μ] t

If s ⊆ t, μ t ≤ μ s, μ t ≠ ∞, and s is measurable, then s =ᵐ[μ] t.

theorem measure_theory.measure_Union_congr_of_subset {α : Type u_1} {β : Type u_2} {m : measurable_space α} {μ : measure_theory.measure α} [countable β] {s t : β set α} (hsub : (b : β), s b t b) (h_le : (b : β), μ (t b) μ (s b)) :
μ ( (b : β), s b) = μ ( (b : β), t b)
theorem measure_theory.measure_union_congr_of_subset {α : Type u_1} {m : measurable_space α} {μ : measure_theory.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 measure_theory.measure_Union_to_measurable {α : Type u_1} {β : Type u_2} {m : measurable_space α} {μ : measure_theory.measure α} [countable β] (s : β set α) :
μ ( (b : β), measure_theory.to_measurable μ (s b)) = μ ( (b : β), s b)
theorem measure_theory.measure_bUnion_to_measurable {α : Type u_1} {β : Type u_2} {m : measurable_space α} {μ : measure_theory.measure α} {I : set β} (hc : I.countable) (s : β set α) :
μ ( (b : β) (H : b I), measure_theory.to_measurable μ (s b)) = μ ( (b : β) (H : b I), s b)
@[simp]
@[simp]
theorem measure_theory.sum_measure_le_measure_univ {α : Type u_1} {ι : Type u_5} {m : measurable_space α} {μ : measure_theory.measure α} {s : finset ι} {t : ι set α} (h : (i : ι), i s measurable_set (t i)) (H : s.pairwise_disjoint t) :
s.sum (λ (i : ι), μ (t i)) μ set.univ
theorem measure_theory.tsum_measure_le_measure_univ {α : Type u_1} {ι : Type u_5} {m : 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} {m : 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} {m : measurable_space α} (μ : measure_theory.measure α) {s : finset ι} {t : ι set α} (h : (i : ι), i s measurable_set (t i)) (H : μ set.univ < s.sum (λ (i : ι), 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.nonempty_inter_of_measure_lt_add {α : Type u_1} {m : measurable_space α} (μ : measure_theory.measure α) {s t u : set α} (ht : measurable_set t) (h's : s u) (h't : t u) (h : μ u < μ s + μ t) :

If two sets s and t are included in a set u, and μ s + μ t > μ u, then s intersects t. Version assuming that t is measurable.

theorem measure_theory.nonempty_inter_of_measure_lt_add' {α : Type u_1} {m : measurable_space α} (μ : measure_theory.measure α) {s t u : set α} (hs : measurable_set s) (h's : s u) (h't : t u) (h : μ u < μ s + μ t) :

If two sets s and t are included in a set u, and μ s + μ t > μ u, then s intersects t. Version assuming that s is measurable.

theorem measure_theory.measure_Union_eq_supr {α : Type u_1} {ι : Type u_5} {m : measurable_space α} {μ : measure_theory.measure α} [countable ι] {s : ι set α} (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 (not necessarily -measurable) sets is the supremum of the measures.

theorem measure_theory.measure_bUnion_eq_supr {α : Type u_1} {ι : Type u_5} {m : measurable_space α} {μ : measure_theory.measure α} {s : ι set α} {t : set ι} (ht : t.countable) (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} {m : measurable_space α} {μ : measure_theory.measure α} [countable 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.tendsto_measure_Union {α : Type u_1} {ι : Type u_5} {m : measurable_space α} {μ : measure_theory.measure α} [semilattice_sup ι] [countable ι] {s : ι set α} (hm : monotone s) :
filter.tendsto (μ s) filter.at_top (nhds (μ ( (n : ι), s n)))

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

theorem measure_theory.tendsto_measure_Inter {α : Type u_1} {ι : Type u_5} {m : measurable_space α} {μ : measure_theory.measure α} [countable ι] [semilattice_sup ι] {s : ι set α} (hs : (n : ι), measurable_set (s n)) (hm : antitone s) (hf : (i : ι), μ (s i) ) :
filter.tendsto (μ s) filter.at_top (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 measure_theory.tendsto_measure_bInter_gt {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {ι : Type u_2} [linear_order ι] [topological_space ι] [order_topology ι] [densely_ordered ι] [topological_space.first_countable_topology ι] {s : ι set α} {a : ι} (hs : (r : ι), r > a measurable_set (s r)) (hm : (i j : ι), a < i i j s i s j) (hf : (r : ι) (H : r > a), μ (s r) ) :
filter.tendsto (μ s) (nhds_within a (set.Ioi a)) (nhds (μ ( (r : ι) (H : 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 measure_theory.measure_limsup_eq_zero {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : ∑' (i : ), μ (s i) ) :

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

theorem measure_theory.measure_liminf_eq_zero {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s : set α} (h : ∑' (i : ), μ (s i) ) :
theorem measure_theory.limsup_ae_eq_of_forall_ae_eq {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} (s : set α) {t : set α} (h : (n : ), s n =ᵐ[μ] t) :
theorem measure_theory.liminf_ae_eq_of_forall_ae_eq {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} (s : set α) {t : set α} (h : (n : ), s n =ᵐ[μ] t) :
theorem measure_theory.measure_if {α : Type u_1} {β : Type u_2} {m : measurable_space α} {μ : measure_theory.measure α} {x : β} {t : set β} {s : set 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.measure_inter_eq_of_measure_eq {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t u : set α} (hs : measurable_set 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 measure_theory.measure.measure_to_measurable_inter {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (ht : μ t ) :

The measurable superset to_measurable μ t of t (which has the same measure as t) satisfies, for any measurable set s, the equality μ (to_measurable μ t ∩ s) = μ (u ∩ s). Here, we require that the measure of t is finite. The conclusion holds without this assumption when the measure is sigma_finite, see measure_to_measurable_inter_of_sigma_finite.

The ℝ≥0∞-module of measures #

@[simp, norm_cast]
theorem measure_theory.measure.coe_zero {α : Type u_1} {m : measurable_space α} :
0 = 0
@[protected, instance]
Equations
@[simp]
@[simp, norm_cast]
theorem measure_theory.measure.coe_add {α : Type u_1} {m : measurable_space α} (μ₁ μ₂ : measure_theory.measure α) :
(μ₁ + μ₂) = μ₁ + μ₂
theorem measure_theory.measure.add_apply {α : Type u_1} {m : measurable_space α} (μ₁ μ₂ : measure_theory.measure α) (s : set α) :
(μ₁ + μ₂) s = μ₁ s + μ₂ s
@[simp, norm_cast]
theorem measure_theory.measure.coe_smul {α : Type u_1} {R : Type u_6} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] {m : measurable_space α} (c : R) (μ : measure_theory.measure α) :
(c μ) = c μ
@[simp]
theorem measure_theory.measure.smul_apply {α : Type u_1} {R : Type u_6} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] {m : measurable_space α} (c : R) (μ : measure_theory.measure α) (s : set α) :
(c μ) s = c μ s
@[simp]
theorem measure_theory.measure.coe_finset_sum {α : Type u_1} {ι : Type u_5} {m : measurable_space α} (I : finset ι) (μ : ι measure_theory.measure α) :
(I.sum (λ (i : ι), μ i)) = I.sum (λ (i : ι), (μ i))
theorem measure_theory.measure.finset_sum_apply {α : Type u_1} {ι : Type u_5} {m : measurable_space α} (I : finset ι) (μ : ι measure_theory.measure α) (s : set α) :
(I.sum (λ (i : ι), μ i)) s = I.sum (λ (i : ι), (μ i) s)
@[simp]
theorem measure_theory.measure.coe_nnreal_smul_apply {α : Type u_1} {m : measurable_space α} (c : nnreal) (μ : measure_theory.measure α) (s : set α) :
(c μ) s = c * μ s
theorem measure_theory.measure.ae_smul_measure_iff {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {p : α Prop} {c : ennreal} (hc : c 0) :
(∀ᵐ (x : α) c μ, p x) ∀ᵐ (x : α) μ, p x
theorem measure_theory.measure.measure_eq_left_of_subset_of_measure_add_eq {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {s t : set α} (h : + ν) t ) (h' : s t) (h'' : + ν) s = + ν) t) :
μ s = μ t
theorem measure_theory.measure.measure_eq_right_of_subset_of_measure_add_eq {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {s t : set α} (h : + ν) t ) (h' : s t) (h'' : + ν) s = + ν) t) :
ν s = ν t
theorem measure_theory.measure.measure_to_measurable_add_inter_left {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (ht : + ν) t ) :
μ (measure_theory.to_measurable + ν) t s) = μ (t s)
theorem measure_theory.measure.measure_to_measurable_add_inter_right {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {s t : set α} (hs : measurable_set s) (ht : + ν) t ) :
ν (measure_theory.to_measurable + ν) t s) = ν (t s)

The complete lattice of measures #

@[protected, 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} {m0 : measurable_space α} {μ₁ μ₂ : measure_theory.measure α} :
μ₁ μ₂ (s : set α), measurable_set s μ₁ s μ₂ s
theorem measure_theory.measure.le_iff' {α : Type u_1} {m0 : 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} {m0 : 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} {m0 : measurable_space α} {μ ν : measure_theory.measure α} :
μ < ν μ ν (s : set α), μ s < ν s
@[protected]
theorem measure_theory.measure.le_add_left {α : Type u_1} {m0 : measurable_space α} {μ ν ν' : measure_theory.measure α} (h : μ ν) :
μ ν' + ν
@[protected]
theorem measure_theory.measure.le_add_right {α : Type u_1} {m0 : measurable_space α} {μ ν ν' : measure_theory.measure α} (h : μ ν) :
μ ν + ν'
@[simp]
theorem measure_theory.measure.top_add {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} :
+ μ =
@[simp]
theorem measure_theory.measure.add_top {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} :
μ + =
@[protected]
theorem measure_theory.measure.zero_le {α : Type u_1} {m0 : 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 as a linear map. It is defined to be 0 if f is not a measurable function.

Equations
@[irreducible]
noncomputable def measure_theory.measure.map {α : Type u_1} {β : Type u_2} [measurable_space β] [measurable_space α] (f : α β) (μ : measure_theory.measure α) :

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

Equations
Instances for measure_theory.measure.map
@[simp]
theorem measure_theory.measure.map_zero {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} [measurable_space β] (f : α β) :
@[protected, simp]
@[protected, simp]
@[simp]
theorem measure_theory.measure.map_apply_of_ae_measurable {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} (hf : ae_measurable f μ) {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.

@[simp]
theorem measure_theory.measure.map_apply {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} (hf : measurable f) {s : set β} (hs : measurable_set s) :
@[simp]
theorem measure_theory.measure.map_id' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} :
measure_theory.measure.map (λ (x : α), x) μ = μ
theorem measure_theory.measure.map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m0 : 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} {m0 : measurable_space α} [measurable_space β] {μ ν : measure_theory.measure α} {f : α β} (h : μ ν) (hf : measurable f) :
theorem measure_theory.measure.le_map_apply {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} (hf : ae_measurable f μ) (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} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} (hf : ae_measurable f μ) {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.

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

If the linearity is not needed, please use comap instead, which works for a larger class of functions.

Equations
theorem measure_theory.measure.comapₗ_apply {α : Type u_1} {s : set α} {β : Type u_2} [measurable_space α] {mβ : measurable_space β} (f : α β) (hfi : function.injective f) (hf : (s : set α), measurable_set s measurable_set (f '' s)) (μ : measure_theory.measure β) (hs : measurable_set s) :
noncomputable def measure_theory.measure.comap {α : Type u_1} {β : Type u_2} [measurable_space β] [measurable_space α] (f : α β) (μ : measure_theory.measure β) :

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

Equations
theorem measure_theory.measure.le_comap_apply {α : Type u_1} {β : Type u_2} [measurable_space α] {mβ : measurable_space β} (f : α β) (μ : measure_theory.measure β) (hfi : function.injective f) (hf : (s : set α), measurable_set s measure_theory.null_measurable_set (f '' s) μ) (s : set α) :
theorem measure_theory.measure.comap_apply {α : Type u_1} {s : set α} {β : Type u_2} [measurable_space α] {mβ : measurable_space β} (f : α β) (hfi : function.injective f) (hf : (s : set α), measurable_set s measurable_set (f '' s)) (μ : measure_theory.measure β) (hs : measurable_set s) :
theorem measure_theory.measure.measure_image_eq_zero_of_comap_eq_zero {α : Type u_1} {β : Type u_2} [measurable_space α] {mβ : measurable_space β} (f : α β) (μ : measure_theory.measure β) (hfi : function.injective f) (hf : (s : set α), measurable_set s measure_theory.null_measurable_set (f '' s) μ) {s : set α} (hs : (measure_theory.measure.comap f μ) s = 0) :
μ (f '' s) = 0
theorem measure_theory.measure.ae_eq_image_of_ae_eq_comap {α : Type u_1} {β : Type u_2} [measurable_space α] {mβ : measurable_space β} (f : α β) (μ : measure_theory.measure β) (hfi : function.injective f) (hf : (s : set α), measurable_set s measure_theory.null_measurable_set (f '' s) μ) {s t : set α} (hst : s =ᵐ[measure_theory.measure.comap f μ] t) :
f '' s =ᵐ[μ] f '' t
theorem measure_theory.measure.comap_preimage {α : Type u_1} {β : Type u_2} [measurable_space α] {mβ : measurable_space β} (f : α β) (μ : measure_theory.measure β) {s : set β} (hf : function.injective f) (hf' : measurable f) (h : (t : set α), measurable_set t measure_theory.null_measurable_set (f '' t) μ) (hs : measurable_set s) :

Subtype of a measure space #

Restricting a measure #

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.

@[simp]
theorem measure_theory.measure.restrict_apply {α : Type u_1} {m0 : 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_mono' {α : Type u_1} {m0 : 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} {m0 : 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_mono_ae {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (h : s ≤ᵐ[μ] t) :
theorem measure_theory.measure.restrict_congr_set {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (h : s =ᵐ[μ] t) :
μ.restrict s = μ.restrict t
@[simp]
theorem measure_theory.measure.restrict_apply' {α : Type u_1} {m0 : 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_le_self {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} :
μ.restrict s μ
theorem measure_theory.measure.restrict_eq_self {α : Type u_1} {m0 : measurable_space α} (μ : measure_theory.measure α) {s t : set α} (h : s t) :
(μ.restrict t) s = μ s
@[simp]
theorem measure_theory.measure.restrict_apply_self {α : Type u_1} {m0 : measurable_space α} (μ : measure_theory.measure α) (s : set α) :
(μ.restrict s) s = μ s
theorem measure_theory.measure.le_restrict_apply {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} (s t : set α) :
μ (t s) (μ.restrict s) t
theorem measure_theory.measure.restrict_apply_superset {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (h : s t) :
(μ.restrict s) t = μ s
@[simp]
theorem measure_theory.measure.restrict_add {α : Type u_1} {m0 : 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} {m0 : measurable_space α} (s : set α) :
0.restrict s = 0
@[simp]
theorem measure_theory.measure.restrict_smul {α : Type u_1} {m0 : measurable_space α} (c : ennreal) (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} {m0 : 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_restrict_of_subset {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (h : s t) :
(μ.restrict t).restrict s = μ.restrict s
theorem measure_theory.measure.restrict_restrict' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (ht : measurable_set t) :
(μ.restrict t).restrict s = μ.restrict (s t)
theorem measure_theory.measure.restrict_comm {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (hs : measurable_set s) :
(μ.restrict t).restrict s = (μ.restrict s).restrict t
theorem measure_theory.measure.restrict_apply_eq_zero {α : Type u_1} {m0 : 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} {m0 : 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} {m0 : 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} {m0 : 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} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (h : μ s = 0) :
μ.restrict s = 0
theorem measure_theory.measure.restrict_inter_add_diff {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {t : set α} (s : set α) (ht : measurable_set t) :
μ.restrict (s t) + μ.restrict (s \ t) = μ.restrict s
theorem measure_theory.measure.restrict_union_add_inter {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {t : set α} (s : set α) (ht : measurable_set t) :
μ.restrict (s t) + μ.restrict (s t) = μ.restrict s + μ.restrict t
theorem measure_theory.measure.restrict_union_add_inter' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) (t : set α) :
μ.restrict (s t) + μ.restrict (s t) = μ.restrict s + μ.restrict t
theorem measure_theory.measure.restrict_union {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (h : disjoint s t) (ht : measurable_set t) :
μ.restrict (s t) = μ.restrict s + μ.restrict t
theorem measure_theory.measure.restrict_union' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (h : disjoint s t) (hs : measurable_set s) :
μ.restrict (s t) = μ.restrict s + μ.restrict t
@[simp]
@[simp]
theorem measure_theory.measure.restrict_union_le {α : Type u_1} {m0 : 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_ae {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable ι] {s : ι set α} (hd : pairwise (measure_theory.ae_disjoint μ on s)) (hm : (i : ι), measure_theory.null_measurable_set (s i) μ) {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 {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable 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} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable ι] {s : ι set α} (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} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} (hf : measurable f) {s : set β} (hs : measurable_set s) :

The restriction of the pushforward measure is the pushforward of the restriction. For a version assuming only ae_measurable, see restrict_map_of_ae_measurable.

theorem measure_theory.measure.restrict_eq_self_of_ae_mem {α : Type u_1} {m0 : measurable_space α} ⦃s : set α⦄ ⦃μ : measure_theory.measure α⦄ (hs : ∀ᵐ (x : α) μ, x s) :
μ.restrict s = μ
theorem measure_theory.measure.restrict_congr_meas {α : Type u_1} {m0 : 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 s measurable_set t μ t = ν t
theorem measure_theory.measure.restrict_congr_mono {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {s t : set α} (hs : s t) (h : μ.restrict t = ν.restrict t) :
μ.restrict s = ν.restrict s
theorem measure_theory.measure.restrict_union_congr {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {s t : set α} :
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} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {s : finset ι} {t : ι set α} :
μ.restrict ( (i : ι) (H : i s), t i) = ν.restrict ( (i : ι) (H : i s), t i) (i : ι), i s μ.restrict (t i) = ν.restrict (t i)
theorem measure_theory.measure.restrict_Union_congr {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ ν : measure_theory.measure α} [countable ι] {s : ι set α} :
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} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {s : set ι} {t : ι set α} (hc : s.countable) :
μ.restrict ( (i : ι) (H : i s), t i) = ν.restrict ( (i : ι) (H : i s), t i) (i : ι), i s μ.restrict (t i) = ν.restrict (t i)
theorem measure_theory.measure.restrict_sUnion_congr {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {S : set (set α)} (hc : S.countable) :
μ.restrict (⋃₀ S) = ν.restrict (⋃₀ S) (s : set α), s S μ.restrict s = ν.restrict s

This lemma shows that Inf and restrict commute for measures.

theorem measure_theory.measure.exists_mem_of_measure_ne_zero_of_ae {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : μ s 0) {p : α Prop} (hp : ∀ᵐ (x : α) μ.restrict s, p x) :
(x : α), x s p x

Extensionality results #

theorem measure_theory.measure.ext_iff_of_Union_eq_univ {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ ν : measure_theory.measure α} [countable ι] {s : ι set α} (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} {m0 : measurable_space α} {μ ν : measure_theory.measure α} [countable ι] {s : ι set α} (hs : ( (i : ι), s i) = set.univ) :
( (i : ι), μ.restrict (s i) = ν.restrict (s i)) μ = ν

Alias of the reverse direction of measure_theory.measure.ext_iff_of_Union_eq_univ.

theorem measure_theory.measure.ext_iff_of_bUnion_eq_univ {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {S : set ι} {s : ι set α} (hc : S.countable) (hs : ( (i : ι) (H : i S), s i) = set.univ) :
μ = ν (i : ι), i S 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 bUnion).

theorem measure_theory.measure.ext_of_bUnion_eq_univ {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {S : set ι} {s : ι set α} (hc : S.countable) (hs : ( (i : ι) (H : i S), s i) = set.univ) :
( (i : ι), i S μ.restrict (s i) = ν.restrict (s i)) μ = ν

Alias of the reverse direction of measure_theory.measure.ext_iff_of_bUnion_eq_univ.

theorem measure_theory.measure.ext_iff_of_sUnion_eq_univ {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {S : set (set α)} (hc : S.countable) (hs : ⋃₀ S = set.univ) :
μ = ν (s : set α), s S μ.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} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {S : set (set α)} (hc : S.countable) (hs : ⋃₀ S = set.univ) :
( (s : set α), s S μ.restrict s = ν.restrict s) μ = ν

Alias of the reverse direction of measure_theory.measure.ext_iff_of_sUnion_eq_univ.

theorem measure_theory.measure.ext_of_generate_from_of_cover {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {S T : set (set α)} (h_gen : m0 = measurable_space.generate_from S) (hc : T.countable) (h_inter : is_pi_system S) (hU : ⋃₀ T = set.univ) (htop : (t : set α), t T μ t ) (ST_eq : (t : set α), t T (s : set α), s S μ (s t) = ν (s t)) (T_eq : (t : set α), t T μ t = ν t) :
μ = ν
theorem measure_theory.measure.ext_of_generate_from_of_cover_subset {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {S T : set (set α)} (h_gen : m0 = measurable_space.generate_from S) (h_inter : is_pi_system S) (h_sub : T S) (hc : T.countable) (hU : ⋃₀ T = set.univ) (htop : (s : set α), s T μ s ) (h_eq : (s : set α), s S μ s = ν s) :
μ = ν

Two measures are equal if they are equal on the π-system generating the σ-algebra, and they are both finite on 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} {m0 : measurable_space α} {μ ν : measure_theory.measure α} (C : set (set α)) (B : set α) (hA : m0 = 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 C μ s = ν s) :
μ = ν

Two measures are equal if they are equal on the π-system generating the σ-algebra, and they are both finite on 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.

noncomputable def measure_theory.measure.dirac {α : Type u_1} [measurable_space α] (a : α) :

The dirac measure.

Equations
Instances for measure_theory.measure.dirac
@[simp]
@[simp]
theorem measure_theory.measure.dirac_apply_of_mem {α : Type u_1} {s : set α} [measurable_space α] {a : α} (h : a s) :
noncomputable def measure_theory.measure.sum {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} (f : ι measure_theory.measure α) :

Sum of an indexed family of measures.

Equations
Instances for measure_theory.measure.sum
theorem measure_theory.measure.le_sum_apply {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} (f : ι measure_theory.measure α) (s : set α) :
@[simp]
theorem measure_theory.measure.sum_apply {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} (f : ι measure_theory.measure α) {s : set α} (hs : measurable_set s) :
theorem measure_theory.measure.le_sum {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} (μ : ι measure_theory.measure α) (i : ι) :
@[simp]
theorem measure_theory.measure.sum_apply_eq_zero {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} [countable ι] {μ : ι measure_theory.measure α} {s : set α} :
(measure_theory.measure.sum μ) s = 0 (i : ι), (μ i) s = 0
theorem measure_theory.measure.sum_apply_eq_zero' {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : ι measure_theory.measure α} {s : set α} (hs : measurable_set s) :
(measure_theory.measure.sum μ) s = 0 (i : ι), (μ i) s = 0
theorem measure_theory.measure.sum_comm {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {ι' : Type u_2} (μ : ι ι' measure_theory.measure α) :
theorem measure_theory.measure.ae_sum_iff {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} [countable ι] {μ : ι measure_theory.measure α} {p : α Prop} :
(∀ᵐ (x : α) measure_theory.measure.sum μ, p x) (i : ι), ∀ᵐ (x : α) μ i, p x
theorem measure_theory.measure.ae_sum_iff' {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : ι measure_theory.measure α} {p : α Prop} (h : measurable_set {x : α | p x}) :
(∀ᵐ (x : α) measure_theory.measure.sum μ, p x) (i : ι), ∀ᵐ (x : α) μ i, p x
@[simp]
theorem measure_theory.measure.sum_fintype {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} [fintype ι] (μ : ι measure_theory.measure α) :
@[simp]
theorem measure_theory.measure.sum_coe_finset {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} (s : finset ι) (μ : ι measure_theory.measure α) :
measure_theory.measure.sum (λ (i : s), μ i) = s.sum (λ (i : ι), μ i)
@[simp]
theorem measure_theory.measure.ae_sum_eq {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} [countable ι] (μ : ι measure_theory.measure α) :
(measure_theory.measure.sum μ).ae = (i : ι), (μ i).ae
@[simp]
theorem measure_theory.measure.sum_cond {α : Type u_1} {m0 : 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} {m0 : measurable_space α} (μ : ι measure_theory.measure α) {s : set α} (hs : measurable_set s) :
@[simp]

If f is a map with countable codomain, then μ.map f is a sum of Dirac measures.

@[simp]

A measure on a countable type is a sum of Dirac measures.

theorem measure_theory.measure.tsum_indicator_apply_singleton {α : Type u_1} {m0 : measurable_space α} [countable α] [measurable_singleton_class α] (μ : measure_theory.measure α) (s : set α) (hs : measurable_set s) :
∑' (x : α), s.indicator (λ (x : α), μ {x}) x = μ s

Given that α is a countable, measurable space with all singleton sets measurable, write the measure of a set s as the sum of the measure of {x} for all x ∈ s.

theorem measure_theory.measure.restrict_Union_ae {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable ι] {s : ι set α} (hd : pairwise (measure_theory.ae_disjoint μ on s)) (hm : (i : ι), measure_theory.null_measurable_set (s i) μ) :
μ.restrict ( (i : ι), s i) = measure_theory.measure.sum (λ (i : ι), μ.restrict (s i))
theorem measure_theory.measure.restrict_Union {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable 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} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable ι] {s : ι set α} :
μ.restrict ( (i : ι), s i) measure_theory.measure.sum (λ (i : ι), μ.restrict (s i))

Counting measure on any measurable space.

Equations
Instances for measure_theory.measure.count

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
Instances for measure_theory.measure.absolutely_continuous
theorem measure_theory.measure.absolutely_continuous.mk {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} (h : ⦃s : set α⦄, measurable_set s ν s = 0 μ s = 0) :
@[protected, trans]
theorem measure_theory.measure.absolutely_continuous.trans {α : Type u_1} {m0 : measurable_space α} {μ₁ μ₂ μ₃ : measure_theory.measure α} (h1 : μ₁.absolutely_continuous μ₂) (h2 : μ₂.absolutely_continuous μ₃) :
theorem measure_theory.measure.absolutely_continuous.ae_eq {α : Type u_1} {δ : Type u_4} {m0 : measurable_space α} {μ ν : measure_theory.measure α} (h : μ.absolutely_continuous 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 β] {m0 : measurable_space α} (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.ae {α : Type u_1} {β : Type u_2} {m0 : 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} {m0 : 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
theorem measure_theory.measure.quasi_measure_preserving.preimage_null {α : Type u_1} {β : Type u_2} {m0 : 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) {s : set β} (hs : μb s = 0) :
μa (f ⁻¹' s) = 0

By replacing a measurable set that is almost invariant with the limsup of its preimages, we obtain a measurable set that is almost equal and strictly invariant.

(The liminf would work just as well.)

The cofinite filter #

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

Equations
theorem measure_theory.measure.eventually_cofinite {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {p : α Prop} :
(∀ᶠ (x : α) in μ.cofinite, p x) μ {x : α | ¬p x} <

The preimage of a null measurable set under a (quasi) measure preserving map is a null measurable set.

@[simp]
theorem measure_theory.ae_eq_bot {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} :
μ.ae = μ = 0
@[simp]
theorem measure_theory.ae_ne_bot {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} :
μ.ae.ne_bot μ 0
@[simp]
theorem measure_theory.ae_zero {α : Type u_1} {m0 : measurable_space α} :
0.ae =
theorem measure_theory.ae_mono {α : Type u_1} {m0 : 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} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} (hf : ae_measurable f μ) {s : set β} (hs : measurable_set s) :
theorem measure_theory.mem_ae_of_mem_ae_map {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} (hf : ae_measurable f μ) {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} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} (hf : ae_measurable f μ) {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} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} (hf : ae_measurable f μ) {p : β Prop} (h : ∀ᵐ (y : β) measure_theory.measure.map f μ, p y) :
∀ᵐ (x : α) μ, p (f x)
@[simp]
theorem measure_theory.ae_restrict_Union_eq {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable ι] (s : ι set α) :
(μ.restrict ( (i : ι), s i)).ae = (i : ι), (μ.restrict (s i)).ae
@[simp]
theorem measure_theory.ae_restrict_union_eq {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} (s t : set α) :
(μ.restrict (s t)).ae = (μ.restrict s).ae (μ.restrict t).ae
theorem measure_theory.ae_restrict_bUnion_eq {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} (s : ι set α) {t : set ι} (ht : t.countable) :
(μ.restrict ( (i : ι) (H : i t), s i)).ae = (i : ι) (H : i t), (μ.restrict (s i)).ae
theorem measure_theory.ae_restrict_bUnion_finset_eq {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} (s : ι set α) (t : finset ι) :
(μ.restrict ( (i : ι) (H : i t), s i)).ae = (i : ι) (H : i t), (μ.restrict (s i)).ae
theorem measure_theory.ae_restrict_Union_iff {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable ι] (s : ι set α) (p : α Prop) :
(∀ᵐ (x : α) μ.restrict ( (i : ι), s i), p x) (i : ι), ∀ᵐ (x : α) μ.restrict (s i), p x
theorem measure_theory.ae_restrict_union_iff {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} (s t : set α) (p : α Prop) :
(∀ᵐ (x : α) μ.restrict (s t), p x) (∀ᵐ (x : α) μ.restrict s, p x) ∀ᵐ (x : α) μ.restrict t, p x
theorem measure_theory.ae_restrict_bUnion_iff {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} (s : ι set α) {t : set ι} (ht : t.countable) (p : α Prop) :
(∀ᵐ (x : α) μ.restrict ( (i : ι) (H : i t), s i), p x) (i : ι), i t (∀ᵐ (x : α) μ.restrict (s i), p x)
@[simp]
theorem measure_theory.ae_restrict_bUnion_finset_iff {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} (s : ι set α) (t : finset ι) (p : α Prop) :
(∀ᵐ (x : α) μ.restrict ( (i : ι) (H : i t), s i), p x) (i : ι), i t (∀ᵐ (x : α) μ.restrict (s i), p x)
theorem measure_theory.ae_eq_restrict_Union_iff {α : Type u_1} {δ : Type u_4} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable ι] (s : ι set α) (f g : α δ) :
f =ᵐ[μ.restrict ( (i : ι), s i)] g (i : ι), f =ᵐ[μ.restrict (s i)] g
theorem measure_theory.ae_eq_restrict_bUnion_iff {α : Type u_1} {δ : Type u_4} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} (s : ι set α) {t : set ι} (ht : t.countable) (f g : α δ) :
f =ᵐ[μ.restrict ( (i : ι) (H : i t), s i)] g (i : ι), i t f =ᵐ[μ.restrict (s i)] g
theorem measure_theory.ae_eq_restrict_bUnion_finset_iff {α : Type u_1} {δ : Type u_4} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} (s : ι set α) (t : finset ι) (f g : α δ) :
f =ᵐ[μ.restrict ( (i : ι) (H : i t), s i)] g (i : ι), i t f =ᵐ[μ.restrict (s i)] g
theorem measure_theory.ae_restrict_uIoc_eq {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [linear_order α] (a b : α) :
(μ.restrict (set.uIoc a b)).ae = (μ.restrict (set.Ioc a b)).ae (μ.restrict (set.Ioc b a)).ae
theorem measure_theory.ae_restrict_uIoc_iff {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [linear_order α] {a b : α} {P : α Prop} :
(∀ᵐ (x : α) μ.restrict (set.uIoc a b), P x) (∀ᵐ (x : α) μ.restrict (set.Ioc a b), P x) ∀ᵐ (x : α) μ.restrict (set.Ioc b a), P x

See also measure_theory.ae_uIoc_iff.

theorem measure_theory.ae_restrict_iff {α : Type u_1} {m0 : 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 s p x
theorem measure_theory.ae_imp_of_ae_restrict {α : Type u_1} {m0 : 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 s p x
theorem measure_theory.ae_restrict_iff' {α : Type u_1} {m0 : 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 s p x
theorem filter.eventually_eq.restrict {α : Type u_1} {δ : Type u_4} {m0 : measurable_space α} {μ : measure_theory.measure α} {f g : α δ} {s : set α} (hfg : f =ᵐ[μ] g) :
theorem measure_theory.ae_restrict_mem {α : Type u_1} {m0 : 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} {m0 : 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_iff'₀ {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} {p : α Prop} (hs : measure_theory.null_measurable_set s μ) :
(∀ᵐ (x : α) μ.restrict s, p x) ∀ᵐ (x : α) μ, x s p x
theorem measure_theory.ae_restrict_of_ae_restrict_of_subset {α : Type u_1} {m0 : 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_of_ae_restrict_of_ae_restrict_compl {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} (t : set α) {p : α Prop} (ht : ∀ᵐ (x : α) μ.restrict t, p x) (htc : ∀ᵐ (x : α) μ.restrict t, p x) :
∀ᵐ (x : α) μ, p x
theorem measure_theory.mem_map_restrict_ae_iff {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {β : Type u_2} {s : set α} {t : set β} {f : α β} (hs : measurable_set s) :
t filter.map f (μ.restrict s).ae μ ((f ⁻¹' t) s) = 0
theorem measure_theory.ae_smul_measure {α : Type u_1} {R : Type u_6} {m0 : measurable_space α} {μ : measure_theory.measure α} {p : α Prop} [monoid R] [distrib_mul_action R ennreal] [is_scalar_tower R ennreal ennreal] (h : ∀ᵐ (x : α) μ, p x) (c : R) :
∀ᵐ (x : α) c μ, p x
theorem measure_theory.ae_add_measure_iff {α : Type u_1} {m0 : 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} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} {f : α β} {g g' : β δ} (hf : ae_measurable f μ) (h : g =ᵐ[ν] g') (h2 : (measure_theory.measure.map f μ).absolutely_continuous ν) :
g f =ᵐ[μ] g' f
theorem measure_theory.measure.quasi_measure_preserving.ae_eq_comp {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} {f : α β} {g g' : β δ} (hf : measure_theory.measure.quasi_measure_preserving f μ ν) (h : g =ᵐ[ν] g') :
g f =ᵐ[μ] g' f
theorem measure_theory.ae_eq_comp {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m0 : measurable_space α} [measurable_space β] {μ : measure_theory.measure α} {f : α β} {g g' : β δ} (hf : ae_measurable f μ) (h : g =ᵐ[measure_theory.measure.map f μ] g') :
g f =ᵐ[μ] g' f
theorem measure_theory.sub_ae_eq_zero {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {β : Type u_2} [add_group β] (f g : α β) :
f - g =ᵐ[μ] 0 f =ᵐ[μ] g
theorem measure_theory.le_ae_restrict {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} :
@[simp]
theorem measure_theory.ae_restrict_eq {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) :
@[simp]
theorem measure_theory.ae_restrict_eq_bot {α : Type u_1} {m0 : 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} {m0 : 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} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) :
s (μ.restrict s).ae
theorem measure_theory.ae_restrict_of_ae_eq_of_ae_restrict {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : α Prop} (hst : s =ᵐ[μ] t) {p : α Prop} :
(∀ᵐ (x : α) μ.restrict s, p x) (∀ᵐ (x : α) μ.restrict t, p x)

If two measurable sets are ae_eq then any proposition that is almost everywhere true on one is almost everywhere true on the other

theorem measure_theory.ae_restrict_congr_set {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : α Prop} (hst : s =ᵐ[μ] t) {p : α Prop} :
(∀ᵐ (x : α) μ.restrict s, p x) ∀ᵐ (x : α) μ.restrict t, p x

If two measurable sets are ae_eq then any proposition that is almost everywhere true on one is almost everywhere true on the other

theorem measure_theory.measure_set_of_frequently_eq_zero {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {p : α Prop} (hp : ∑' (i : ), μ {x : α | p i x} ) :
μ {x : α | ∃ᶠ (n : ) in filter.at_top, p n x} = 0

A version of the Borel-Cantelli lemma: if pᵢ is a sequence of predicates such that ∑ μ {x | pᵢ x} is finite, then the measure of x such that pᵢ x holds frequently as i → ∞ (or equivalently, pᵢ x holds for infinitely many i) is equal to zero.

theorem measure_theory.ae_eventually_not_mem {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (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 sets such that ∑ μ sᵢ exists, then for almost all x, x does not belong to almost all sᵢ.

theorem measure_theory.bsupr_measure_Iic {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [preorder α] {s : set α} (hsc : s.countable) (hst : (x : α), (y : α) (H : y s), x y) (hdir : directed_on has_le.le s) :
( (x : α) (H : x s), μ (set.Iic x)) = μ set.univ
theorem measure_theory.Iio_ae_eq_Iic' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [partial_order α] {a : α} (ha : μ {a} = 0) :
theorem measure_theory.Ioi_ae_eq_Ici' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [partial_order α] {a : α} (ha : μ {a} = 0) :
theorem measure_theory.Ioo_ae_eq_Ioc' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [partial_order α] {a b : α} (hb : μ {b} = 0) :
theorem measure_theory.Ioc_ae_eq_Icc' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [partial_order α] {a b : α} (ha : μ {a} = 0) :
theorem measure_theory.Ioo_ae_eq_Ico' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [partial_order α] {a b : α} (ha : μ {a} = 0) :
theorem measure_theory.Ioo_ae_eq_Icc' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [partial_order α] {a b : α} (ha : μ {a} = 0) (hb : μ {b} = 0) :
theorem measure_theory.Ico_ae_eq_Icc' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [partial_order α] {a b : α} (hb : μ {b} = 0) :
theorem measure_theory.Ico_ae_eq_Ioc' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [partial_order α] {a b : α} (ha : μ {a} = 0) (hb : μ {b} = 0) :
theorem measure_theory.mem_ae_dirac_iff {α : Type u_1} {s : set α} [measurable_space 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}) :
@[class]
structure measure_theory.is_finite_measure {α : Type u_1} {m0 : measurable_space α} (μ : measure_theory.measure α) :
Prop

A measure μ is called finite if μ univ < ∞.

Instances of this typeclass
@[protected, instance]
theorem measure_theory.measure_compl_le_add_of_le_add {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} [measure_theory.is_finite_measure μ] (hs : measurable_set s) (ht : measurable_set t) {ε : ennreal} (h : μ s μ t + ε) :
μ t μ s + ε
theorem measure_theory.measure_compl_le_add_iff {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} [measure_theory.is_finite_measure μ] (hs : measurable_set s) (ht : measurable_set t) {ε : ennreal} :
μ s μ t + ε μ t μ s + ε

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} {m0 : measurable_space α} {μ ν₁ ν₂ : measure_theory.measure α} [measure_theory.is_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} {m0 : measurable_space α} {μ : measure_theory.measure α} [hμ : measure_theory.is_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)
theorem measure_theory.ae_iff_measure_eq {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [measure_theory.is_finite_measure μ] {p : α Prop} (hp : measure_theory.null_measurable_set {a : α | p a} μ) :
(∀ᵐ (a : α) μ, p a) μ {a : α | p a} = μ set.univ
@[simp]

Note that this is not quite as useful as it looks because the measure takes values in ℝ≥0∞. Thus the subtraction appearing is the truncated subtraction of ℝ≥0∞, rather than the better-behaved subtraction of .

@[class]
structure measure_theory.has_no_atoms {α : Type u_1} {m0 : 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 of this typeclass
theorem set.countable.measure_zero {α : Type u_1} {m : measurable_space α} {s : set α} (h : s.countable) (μ : measure_theory.measure α) [measure_theory.has_no_atoms μ] :
μ s = 0
theorem set.countable.ae_not_mem {α : Type u_1} {m : measurable_space α} {s : set α} (h : s.countable) (μ : measure_theory.measure α) [measure_theory.has_no_atoms μ] :
∀ᵐ (x : α) μ, x s
theorem set.finite.measure_zero {α : Type u_1} {m : measurable_space α} {s : set α} (h : s.finite) (μ : measure_theory.measure α) [measure_theory.has_no_atoms μ] :
μ s = 0
theorem measure_theory.ite_ae_eq_of_measure_zero {α : Type u_1} {m0 : 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} {m0 : 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} {m0 : 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.small_sets.

Equations
theorem measure_theory.measure.finite_at_filter.exists_mem_basis {α : Type u_1} {ι : Type u_5} {m0 : 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} {m0 : 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.

Instances for measure_theory.measure.finite_spanning_sets_in
  • measure_theory.measure.finite_spanning_sets_in.has_sizeof_inst

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
noncomputable def measure_theory.spanning_sets_index {α : Type u_1} {m0 : measurable_space α} (μ : measure_theory.measure α) [measure_theory.sigma_finite μ] (x : α) :

spanning_sets_index μ x is the least n : ℕ such that x ∈ spanning_sets μ n.

Equations
theorem measure_theory.measure.exists_subset_measure_lt_top {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} [measure_theory.sigma_finite μ] {r : ennreal} (hs : measurable_set s) (h's : r < μ s) :
(t : set α), measurable_set t t s r < μ t μ t <

In a σ-finite space, any measurable set of measure > r contains a measurable subset of finite measure > r.

A set in a σ-finite space has zero measure if and only if its intersection with all members of the countable family of finite measure spanning sets has zero measure.

A set in a σ-finite space has positive measure if and only if its intersection with some member of the countable family of finite measure spanning sets has positive measure.

theorem measure_theory.measure.finite_const_le_meas_of_disjoint_Union {α : Type u_1} {ι : Type u_2} [measurable_space α] (μ : measure_theory.measure α) {ε : ennreal} (ε_pos : 0 < ε) {As : ι set α} (As_mble : (i : ι), measurable_set (As i)) (As_disj : pairwise (disjoint on As)) (Union_As_finite : μ ( (i : ι), As i) ) :
{i : ι | ε μ (As i)}.finite

If the union of disjoint measurable sets has finite measure, then there are only finitely many members of the union whose measure exceeds any given positive number.

theorem measure_theory.measure.countable_meas_pos_of_disjoint_of_meas_Union_ne_top {α : Type u_1} {ι : Type u_2} [measurable_space α] (μ : measure_theory.measure α) {As : ι set α} (As_mble : (i : ι), measurable_set (As i)) (As_disj : pairwise (disjoint on As)) (Union_As_finite : μ ( (i : ι), As i) ) :
{i : ι | 0 < μ (As i)}.countable

If the union of disjoint measurable sets has finite measure, then there are only countably many members of the union whose measure is positive.

theorem measure_theory.measure.countable_meas_pos_of_disjoint_Union {α : Type u_1} {ι : Type u_2} [measurable_space α] {μ : measure_theory.measure α} [measure_theory.sigma_finite μ] {As : ι set α} (As_mble : (i : ι), measurable_set (As i)) (As_disj : pairwise (disjoint on As)) :
{i : ι | 0 < μ (As i)}.countable

In a σ-finite space, among disjoint measurable sets, only countably many can have positive measure.

theorem measure_theory.measure.countable_meas_level_set_pos {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} [measure_theory.sigma_finite μ] [measurable_space β] [measurable_singleton_class β] {g : α β} (g_mble : measurable g) :
{t : β | 0 < μ {a : α | g a = t}}.countable
theorem measure_theory.measure.measure_to_measurable_inter_of_cover {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) {t : set α} {v : set α} (hv : t (n : ), v n) (h'v : (n : ), μ (t v n) ) :

If a set t is covered by a countable family of finite measure sets, then its measurable superset to_measurable μ t (which has the same measure as t) satisfies, for any measurable set s, the equality μ (to_measurable μ t ∩ s) = μ (t ∩ s).

theorem measure_theory.measure.restrict_to_measurable_of_cover {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} {v : set α} (hv : s (n : ), v n) (h'v : (n : ), μ (s v n) ) :

The measurable superset to_measurable μ t of t (which has the same measure as t) satisfies, for any measurable set s, the equality μ (to_measurable μ t ∩ s) = μ (t ∩ s). This only holds when μ is σ-finite. For a version without this assumption (but requiring that t has finite measure), see measure_to_measurable_inter.

@[protected]
def measure_theory.measure.finite_spanning_sets_in.mono' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {C D : set (set α)} (h : μ.finite_spanning_sets_in C) (hC : C {s : set α | μ s < } D) :

If μ has finite spanning sets in C and C ∩ {s | μ s < ∞} ⊆ D then μ has finite spanning sets in D.

Equations
@[protected]

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

Equations
@[protected]

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

@[protected]
theorem measure_theory.measure.finite_spanning_sets_in.ext {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {C : set (set α)} (hA : m0 = measurable_space.generate_from C) (hC : is_pi_system C) (h : μ.finite_spanning_sets_in C) (h_eq : (s : set α), s C μ 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} {m0 : measurable_space α} {μ : measure_theory.measure α} {S : set (set α)} (hc : S.countable) (hμ : (s : set α), s S μ s < ) (hU : ⋃₀ S = set.univ) :

Given measures μ, ν where ν ≤ μ, finite_spanning_sets_in.of_le provides the induced finite_spanning_set with respect to ν from a finite_spanning_set with respect to μ.

Equations
@[protected, instance]

Every finite measure is σ-finite.

@[protected, instance]

Similar to ae_of_forall_measure_lt_top_ae_restrict, but where you additionally get the hypothesis that another σ-finite measure has finite values on s.

theorem measure_theory.ae_of_forall_measure_lt_top_ae_restrict {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [measure_theory.sigma_finite μ] (P : α Prop) (h : (s : set α), measurable_set s μ s < (∀ᵐ (x : α) μ.restrict s, P x)) :
∀ᵐ (x : α) μ, P x

To prove something for almost all x w.r.t. a σ-finite measure, it is sufficient to show that this holds almost everywhere in sets where the measure has finite value.

A compact subset has finite measure for a measure which is finite on compacts.

A bounded subset has finite measure for a measure which is finite on compact sets, in a proper space.

A measure which is finite on compact sets in a locally compact space is locally finite. Not registered as an instance to avoid a loop with the other direction.

theorem measure_theory.exists_pos_measure_of_cover {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} {μ : measure_theory.measure α} [countable ι] {U : ι set α} (hU : ( (i : ι), U i) = set.univ) (hμ : μ 0) :
(i : ι), 0 < μ (U i)
theorem measure_theory.exists_pos_preimage_ball {α : Type u_1} {δ : Type u_4} {m0 : measurable_space α} {μ : measure_theory.measure α} [pseudo_metric_space δ] (f : α δ) (x : δ) (hμ : μ 0) :
(n : ), 0 < μ (f ⁻¹' metric.ball x n)
theorem measure_theory.exists_pos_ball {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [pseudo_metric_space α] (x : α) (hμ : μ 0) :
(n : ), 0 < μ (metric.ball x n)
theorem measure_theory.null_of_locally_null {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} [topological_space α] [topological_space.second_countable_topology α] (s : set α) (hs : (x : α), x s ( (u : set α) (H : u nhds_within x s), μ u = 0)) :
μ s = 0

If a set has zero measure in a neighborhood of each of its points, then it has zero measure in a second-countable space.

theorem measure_theory.exists_ne_forall_mem_nhds_pos_measure_preimage {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {β : Type u_2} [topological_space β] [t1_space β] [topological_space.second_countable_topology β] [nonempty β] {f : α β} (h : (b : β), ∃ᵐ (x : α) μ, f x b) :
(a b : β), a b ( (s : set β), s nhds a 0 < μ (f ⁻¹' s)) (t : set β), t nhds b 0 < μ (f ⁻¹' t)
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.is_finite_measure μ] (C : set (set α)) (hμν : (s : set α), s C μ 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 : 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} {m0 : measurable_space α} {μ ν : measure_theory.measure α} (C : set (set α)) (hA : m0 = measurable_space.generate_from C) (hC : is_pi_system C) [measure_theory.is_finite_measure μ] (hμν : (s : set α), s C μ 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).

@[protected]

Given S : μ.finite_spanning_sets_in {s | measurable_set s}, finite_spanning_sets_in.disjointed provides a finite_spanning_sets_in {s | measurable_set s} such that its underlying sets are pairwise disjoint.

Equations
@[protected]
@[protected]
theorem measure_theory.measure.finite_at_filter.mono {α : Type u_1} {m0 : measurable_space α} {μ ν : measure_theory.measure α} {f g : filter α} (hf : f g) (hμ : μ ν) :
@[protected]
theorem measurable_embedding.map_apply {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {m1 : measurable_space β} {f : α β} (hf : measurable_embedding f) (μ : measure_theory.measure α) (s : set β) :
theorem measurable_embedding.comap_apply {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {m1 : measurable_space β} {f : α β} (hf : measurable_embedding f) (μ : measure_theory.measure β) (s : set α) :
theorem measurable_embedding.ae_map_iff {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {m1 : measurable_space β} {f : α β} (hf : measurable_embedding f) {p : β Prop} {μ : measure_theory.measure α} :
(∀ᵐ (x : β) measure_theory.measure.map f μ, p x) ∀ᵐ (x : α) μ, p (f x)
@[protected]
theorem measurable_embedding.comap_preimage {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {m1 : measurable_space β} {f : α β} (hf : measurable_embedding f) (μ : measure_theory.measure β) {s : set β} (hs : measurable_set s) :
theorem comap_subtype_coe_apply {α : Type u_1} {m0 : measurable_space α} {s : set α} (hs : measurable_set s) (μ : measure_theory.measure α) (t : set s) :
theorem ae_restrict_iff_subtype {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) {p : α Prop} :

Volume on s : set α #

Interactions of measurable equivalences and measures

@[protected]
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).

noncomputable def measure_theory.measure.trim {α : Type u_1} {m m0 : measurable_space α} (μ : measure_theory.measure α) (hm : m m0) :

Restriction of a measure to a sub-sigma algebra. It is common to see a measure μ on a measurable space structure m0 as being also a measure on any m ≤ m0. Since measures in mathlib have to be trimmed to the measurable space, μ itself cannot be a measure on m, hence the definition of μ.trim hm.

This notion is related to outer_measure.trim, see the lemma to_outer_measure_trim_eq_trim_to_outer_measure.

Equations
Instances for measure_theory.measure.trim
@[simp]
@[simp]
theorem measure_theory.zero_trim {α : Type u_1} {m m0 : measurable_space α} (hm : m m0) :
0.trim hm = 0
theorem measure_theory.trim_measurable_set_eq {α : Type u_1} {m m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hm : m m0) (hs : measurable_set s) :
(μ.trim hm) s = μ s
theorem measure_theory.le_trim {α : Type u_1} {m m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hm : m m0) :
μ s (μ.trim hm) s
theorem measure_theory.measure_eq_zero_of_trim_eq_zero {α : Type u_1} {m m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hm : m m0) (h : (μ.trim hm) s = 0) :
μ s = 0
theorem measure_theory.measure_trim_to_measurable_eq_zero {α : Type u_1} {m m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} {hm : m m0} (hs : (μ.trim hm) s = 0) :
theorem measure_theory.ae_of_ae_trim {α : Type u_1} {m m0 : measurable_space α} (hm : m m0) {μ : measure_theory.measure α} {P : α Prop} (h : ∀ᵐ (x : α) μ.trim hm, P x) :
∀ᵐ (x : α) μ, P x
theorem measure_theory.ae_eq_of_ae_eq_trim {α : Type u_1} {m m0 : measurable_space α} {μ : measure_theory.measure α} {E : Type u_2} {hm : m m0} {f₁ f₂ : α E} (h12 : f₁ =ᵐ[μ.trim hm] f₂) :
f₁ =ᵐ[μ] f₂
theorem measure_theory.ae_le_of_ae_le_trim {α : Type u_1} {m m0 : measurable_space α} {μ : measure_theory.measure α} {E : Type u_2} [has_le E] {hm : m m0} {f₁ f₂ : α E} (h12 : f₁ ≤ᵐ[μ.trim hm] f₂) :
f₁ ≤ᵐ[μ] f₂
theorem measure_theory.trim_trim {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {m₁ m₂ : measurable_space α} {hm₁₂ : m₁ m₂} {hm₂ : m₂ m0} :
(μ.trim hm₂).trim hm₁₂ = μ.trim _
theorem measure_theory.restrict_trim {α : Type u_1} {m m0 : measurable_space α} {s : set α} (hm : m m0) (μ : measure_theory.measure α) (hs : measurable_set s) :
(μ.trim hm).restrict s = (μ.restrict s).trim hm
theorem measure_theory.sigma_finite_trim_mono {α : Type u_1} {m m₂ m0 : measurable_space α} {μ : measure_theory.measure α} (hm : m m0) (hm₂ : m₂ m) [measure_theory.sigma_finite (μ.trim _)] :
theorem is_compact.exists_open_superset_measure_lt_top' {α : Type u_1} [topological_space α] [measurable_space α] {μ : measure_theory.measure α} {s : set α} (h : is_compact s) (hμ : (x : α), x s μ.finite_at_filter (nhds x)) :
(U : set α) (H : U s), is_open U μ U <

If s is a compact set and μ is finite at 𝓝 x for every x ∈ s, then s admits an open superset of finite measure.

If s is a compact set and μ is a locally finite measure, then s admits an open superset of finite measure.

theorem is_compact.measure_lt_top_of_nhds_within {α : Type u_1} [topological_space α] [measurable_space α] {μ : measure_theory.measure α} {s : set α} (h : is_compact s) (hμ : (x : α), x s μ.finite_at_filter (nhds_within x s)) :
μ s <
theorem is_compact.measure_zero_of_nhds_within {α : Type u_1} [topological_space α] [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : is_compact s) :
( (a : α), a s ( (t : set α) (H : t nhds_within a s), μ t = 0)) μ s = 0

A locally finite measure on a σ-compact topological space admits a finite spanning sequence of open sets.

Equations
@[irreducible]

A locally finite measure on a second countable topological space admits a finite spanning sequence of open sets.

Equations
theorem piecewise_ae_eq_restrict {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f g : α β} (hs : measurable_set s) :
theorem piecewise_ae_eq_restrict_compl {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f g : α β} (hs : measurable_set s) :
theorem piecewise_ae_eq_of_ae_eq_set {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} {f g : α β} (hst : s =ᵐ[μ] t) :
s.piecewise f g =ᵐ[μ] t.piecewise f g
theorem mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f : α β} [has_zero β] {t : set β} (ht : 0 t) (hs : measurable_set s) :
theorem mem_map_indicator_ae_iff_of_zero_nmem {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f : α β} [has_zero β] {t : set β} (ht : 0 t) :
t filter.map (s.indicator f) μ.ae μ ((f ⁻¹' t) s) = 0
theorem map_restrict_ae_le_map_indicator_ae {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f : α β} [has_zero β] (hs : measurable_set s) :
theorem indicator_ae_eq_restrict {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f : α β} [has_zero β] (hs : measurable_set s) :
theorem indicator_ae_eq_restrict_compl {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f : α β} [has_zero β] (hs : measurable_set s) :
theorem indicator_ae_eq_of_restrict_compl_ae_eq_zero {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f : α β} [has_zero β] (hs : measurable_set s) (hf : f =ᵐ[μ.restrict s] 0) :
theorem indicator_ae_eq_zero_of_restrict_ae_eq_zero {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f : α β} [has_zero β] (hs : measurable_set s) (hf : f =ᵐ[μ.restrict s] 0) :
theorem indicator_ae_eq_of_ae_eq_set {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s t : set α} {f : α β} [has_zero β] (hst : s =ᵐ[μ] t) :
theorem indicator_meas_zero {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f : α β} [has_zero β] (hs : μ s = 0) :
theorem ae_eq_restrict_iff_indicator_ae_eq {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f : α β} [has_zero β] {g : α β} (hs : measurable_set s) :