# mathlib3documentation

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:

• `is_probability_measure μ`: `μ univ = 1`;
• `is_finite_measure μ`: `μ univ < ∞`;
• `sigma_finite μ`: there exists a countable collection of sets that cover `univ` where `μ` is finite;
• `is_locally_finite_measure μ` : `∀ x, ∃ s ∈ 𝓝 x, μ s < ∞`;
• `has_no_atoms μ` : `∀ x, μ {x} = 0`; possibly should be redefined as `∀ s, 0 < μ s → ∃ t ⊆ s, 0 < μ t ∧ μ t < μ s`.

Given a measure, the null sets are the sets where `μ s = 0`, where `μ` denotes the corresponding outer measure (so `s` might not be measurable). We can then define the completion of `μ` as the measure on the least `σ`-algebra that also contains all null sets, by defining the measure to be `0` on the null sets.

## Main statements #

• `completion` is the completion of a measure to all null measurable sets.
• `measure.of_measurable` and `outer_measure.to_measure` are two important ways to define a measure.

## Implementation notes #

Given `μ : measure α`, `μ s` is the value of the outer measure applied to `s`. This conveniently allows us to apply the measure to sets without proving that they are measurable. We get countable subadditivity for all sets, but only countable additivity for measurable sets.

You often don't want to define a measure via its constructor. Two ways that are sometimes more convenient:

• `measure.of_measurable` is a way to define a measure by only giving its value on measurable sets and proving the properties (1) and (2) mentioned above.
• `outer_measure.to_measure` is a way of obtaining a measure from an outer measure by showing that all measurable sets in the measurable space are Carathéodory measurable.

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

• `ext`: two measures are equal if they are equal on all measurable sets.
• `ext_of_generate_from_of_Union`: two measures are equal if they are equal on a π-system generating the measurable sets, if the π-system contains a spanning increasing sequence of sets where the measures take finite value (in particular the measures are σ-finite). This is a special case of the more general `ext_of_generate_from_of_cover`
• `ext_of_generate_finite`: two finite measures are equal if they are equal on a π-system generating the measurable sets. This is a special case of `ext_of_generate_from_of_Union` using `C ∪ {univ}`, but is easier to work with.

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

## Tags #

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

@[protected, instance]
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 b P x) (∀ᵐ (x : α) μ, x b P x) ∀ᵐ (x : α) μ, x 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_add_measure_compl {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s : set α} (h : measurable_set s) :
μ s + μ s =
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 ) (h : (b : β), b s ) :
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 ) (h : (s : set α), ) :
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 ) :
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 ) (hm : (b : ι), b s ) :
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} (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 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), (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), (s b)) = μ ( (b : β) (H : b I), s b)
@[simp]
theorem measure_theory.measure_to_measurable_union {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s t : set α} :
μ = μ (s t)
@[simp]
theorem measure_theory.measure_union_to_measurable {α : Type u_1} {m : measurable_space α} {μ : measure_theory.measure α} {s t : set α} :
μ = μ (s t)
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))
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)
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 : < ∑' (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 : < 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 : 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 : 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 : 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} [countable ι] {s : ι set α} (hm : monotone s) :
(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] {s : ι set α} (hs : (n : ι), measurable_set (s n)) (hm : antitone s) (hf : (i : ι), μ (s i) ) :
(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] {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) (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) ) :
= 0

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

theorem 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) ) :
= 0
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
noncomputable def measure_theory.outer_measure.to_measure {α : Type u_1} [ms : measurable_space α] (h : ms m.caratheodory) :

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] (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] (h : ms m.caratheodory) (s : set α) :
m s (m.to_measure h) s
theorem measure_theory.to_measure_apply₀ {α : Type u_1} [ms : measurable_space α] (h : ms m.caratheodory) {s : set α} (hs : (m.to_measure h)) :
(m.to_measure h) s = m s
@[simp]
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 ) :
μ = μ (t s)

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 #

@[protected, instance]
Equations
@[simp, norm_cast]
theorem measure_theory.measure.coe_zero {α : Type u_1} {m : measurable_space α} :
0 = 0
@[protected, instance]
@[protected, instance]
Equations
@[protected, instance]
noncomputable def measure_theory.measure.has_add {α : Type u_1}  :
Equations
@[simp]
theorem measure_theory.measure.add_to_outer_measure {α : Type u_1} {m : measurable_space α} (μ₁ μ₂ : measure_theory.measure α) :
(μ₁ + μ₂).to_outer_measure =
@[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
@[protected, instance]
def measure_theory.measure.has_smul {α : Type u_1} {R : Type u_6} [ ennreal]  :
Equations
@[simp]
theorem measure_theory.measure.smul_to_outer_measure {α : Type u_1} {R : Type u_6} [ ennreal] {m : measurable_space α} (c : R) (μ : measure_theory.measure α) :
@[simp, norm_cast]
theorem measure_theory.measure.coe_smul {α : Type u_1} {R : Type u_6} [ 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} [ ennreal] {m : measurable_space α} (c : R) (μ : measure_theory.measure α) (s : set α) :
(c μ) s = c μ s
@[protected, instance]
def measure_theory.measure.smul_comm_class {α : Type u_1} {R : Type u_6} {R' : Type u_7} [ ennreal] [has_smul R' ennreal] [ R' ennreal]  :
@[protected, instance]
def measure_theory.measure.is_scalar_tower {α : Type u_1} {R : Type u_6} {R' : Type u_7} [ ennreal] [has_smul R' ennreal] [ R'] [ R' ennreal]  :
@[protected, instance]
def measure_theory.measure.is_central_scalar {α : Type u_1} {R : Type u_6} [ ennreal]  :
@[protected, instance]
def measure_theory.measure.mul_action {α : Type u_1} {R : Type u_6} [monoid R]  :
Equations
@[protected, instance]
noncomputable def measure_theory.measure.add_comm_monoid {α : Type u_1}  :
Equations

Coercion to function as an additive monoid homomorphism.

Equations
@[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 ) :
(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 ) (s : set α) :
(I.sum (λ (i : ι), μ i)) s = I.sum (λ (i : ι), (μ i) s)
@[protected, instance]
noncomputable def measure_theory.measure.distrib_mul_action {α : Type u_1} {R : Type u_6} [monoid R]  :
Equations
@[protected, instance]
noncomputable def measure_theory.measure.module {α : Type u_1} {R : Type u_6} [semiring R] [ ennreal]  :
Equations
@[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 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 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), μ₁ s μ₂ s
theorem measure_theory.measure.to_outer_measure_le {α : Type u_1} {m0 : measurable_space α} {μ₁ μ₂ : measure_theory.measure α} :
μ₁ μ₂
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), 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, instance]
@[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'
theorem measure_theory.measure.Inf_caratheodory {α : Type u_1} {m0 : measurable_space α} {m : set } (s : set α) (hs : measurable_set s) :
@[protected, instance]
noncomputable def measure_theory.measure.has_Inf {α : Type u_1}  :
Equations
theorem measure_theory.measure.Inf_apply {α : Type u_1} {m0 : measurable_space α} {s : set α} {m : set } (hs : measurable_set s) :
@[protected, instance]
noncomputable def measure_theory.measure.complete_semilattice_Inf {α : Type u_1}  :
Equations
@[protected, instance]
noncomputable def measure_theory.measure.complete_lattice {α : Type u_1}  :
Equations
@[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]
@[simp]

### Pushforward and pullback #

noncomputable def measure_theory.measure.lift_linear {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} (hf : (μ : , _inst_1 ) :

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
@[simp]
theorem measure_theory.measure.lift_linear_apply {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {μ : measure_theory.measure α} (hf : (μ : , _inst_1 ) {s : set β} (hs : measurable_set s) :
theorem measure_theory.measure.le_lift_linear_apply {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {μ : measure_theory.measure α} (hf : (μ : , _inst_1 ) (s : set β) :
noncomputable def measure_theory.measure.mapₗ {α : Type u_1} {β : Type u_2} (f : α β) :

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

Equations
theorem measure_theory.measure.mapₗ_congr {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {μ : measure_theory.measure α} {f g : α β} (hf : measurable f) (hg : measurable g) (h : f =ᵐ[μ] g) :
@[irreducible]
noncomputable def measure_theory.measure.map {α : Type u_1} {β : Type u_2} (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
• = dite μ) (λ (hf : μ), μ) (λ (hf : ¬ μ), 0)
Instances for `measure_theory.measure.map`
theorem measure_theory.measure.mapₗ_mk_apply_of_ae_measurable {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {μ : measure_theory.measure α} {f : α β} (hf : μ) :
theorem measure_theory.measure.mapₗ_apply_of_measurable {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {f : α β} (hf : measurable f) (μ : measure_theory.measure α) :
@[simp]
theorem measure_theory.measure.map_add {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} (μ ν : measure_theory.measure α) {f : α β} (hf : measurable f) :
@[simp]
theorem measure_theory.measure.map_zero {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} (f : α β) :
theorem measure_theory.measure.map_of_not_ae_measurable {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {f : α β} {μ : measure_theory.measure α} (hf : ¬) :
theorem measure_theory.measure.map_congr {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {μ : measure_theory.measure α} {f g : α β} (h : f =ᵐ[μ] g) :
@[protected, simp]
theorem measure_theory.measure.map_smul {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} (c : ennreal) (μ : measure_theory.measure α) (f : α β) :
(c μ) =
@[protected, simp]
theorem measure_theory.measure.map_smul_nnreal {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} (c : nnreal) (μ : measure_theory.measure α) (f : α β) :
(c μ) =
@[simp]
theorem measure_theory.measure.map_apply_of_ae_measurable {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {μ : measure_theory.measure α} {f : α β} (hf : μ) {s : set β} (hs : measurable_set s) :
s = μ (f ⁻¹' 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} {μ : measure_theory.measure α} {f : α β} (hf : measurable f) {s : set β} (hs : measurable_set s) :
s = μ (f ⁻¹' s)
theorem measure_theory.measure.map_to_outer_measure {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {μ : measure_theory.measure α} {f : α β} (hf : μ) :
@[simp]
@[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} {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} {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} {μ : measure_theory.measure α} {f : α β} (hf : μ) (s : set β) :
μ (f ⁻¹' s) s

Even if `s` is not measurable, we can bound `map f μ s` from below. See also `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} {μ : measure_theory.measure α} {f : α β} (hf : μ) {s : set β} (hs : s = 0) :
μ (f ⁻¹' s) = 0

Even if `s` is not measurable, `map f μ s = 0` implies that `μ (f ⁻¹' s) = 0`.

theorem measure_theory.measure.tendsto_ae_map {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} {μ : measure_theory.measure α} {f : α β} (hf : μ) :
μ.ae .ae
noncomputable def measure_theory.measure.comapₗ {α : Type u_1} {β : Type u_2} (f : α β) :

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

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

Equations
theorem measure_theory.measure.comapₗ_apply {α : Type u_1} {s : set α} {β : Type u_2} {mβ : measurable_space β} (f : α β) (hfi : function.injective f) (hf : (s : set α), measurable_set (f '' s)) (μ : measure_theory.measure β) (hs : measurable_set s) :
s = μ (f '' s)
noncomputable def measure_theory.measure.comap {α : Type u_1} {β : Type u_2} (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.comap_apply₀ {α : Type u_1} {β : Type u_2} {s : set α} (f : α β) (μ : measure_theory.measure β) (hfi : function.injective f) (hf : (s : set α), )  :
= μ (f '' s)
theorem measure_theory.measure.le_comap_apply {α : Type u_1} {β : Type u_2} {mβ : measurable_space β} (f : α β) (μ : measure_theory.measure β) (hfi : function.injective f) (hf : (s : set α), ) (s : set α) :
μ (f '' s)
theorem measure_theory.measure.comap_apply {α : Type u_1} {s : set α} {β : Type u_2} {mβ : measurable_space β} (f : α β) (hfi : function.injective f) (hf : (s : set α), measurable_set (f '' s)) (μ : measure_theory.measure β) (hs : measurable_set s) :
= μ (f '' s)
theorem measure_theory.measure.comapₗ_eq_comap {α : Type u_1} {s : set α} {β : Type u_2} {mβ : measurable_space β} (f : α β) (hfi : function.injective f) (hf : (s : set α), measurable_set (f '' s)) (μ : measure_theory.measure β) (hs : measurable_set s) :
s =
theorem measure_theory.measure.measure_image_eq_zero_of_comap_eq_zero {α : Type u_1} {β : Type u_2} {mβ : measurable_space β} (f : α β) (μ : measure_theory.measure β) (hfi : function.injective f) (hf : (s : set α), ) {s : set α} (hs : = 0) :
μ (f '' s) = 0
theorem measure_theory.measure.ae_eq_image_of_ae_eq_comap {α : Type u_1} {β : Type u_2} {mβ : measurable_space β} (f : α β) (μ : measure_theory.measure β) (hfi : function.injective f) (hf : (s : set α), ) {s t : set α} (hst : s =ᵐ[] t) :
f '' s =ᵐ[μ] f '' t
theorem measure_theory.measure.null_measurable_set.image {α : Type u_1} {β : Type u_2} {mβ : measurable_space β} (f : α β) (μ : measure_theory.measure β) (hfi : function.injective f) (hf : (s : set α), ) {s : set α}  :
theorem measure_theory.measure.comap_preimage {α : Type u_1} {β : Type u_2} {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), ) (hs : measurable_set s) :
(f ⁻¹' s) = μ (s

### Subtype of a measure space #

theorem measure_theory.measure.null_measurable_set.subtype_coe {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} {t : set s} (hs : μ)  :
theorem measure_theory.measure.measure_subtype_coe_le_comap {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : μ) (t : set s) :
μ (coe '' t)
theorem measure_theory.measure.measure_subtype_coe_eq_zero_of_comap_eq_zero {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (hs : μ) {t : set s} (ht : = 0) :
μ (coe '' t) = 0
@[protected, instance]
noncomputable def measure_theory.measure.subtype.measure_space {α : Type u_1} {p : α Prop} :
Equations

### Restricting a measure #

noncomputable def measure_theory.measure.restrictₗ {α : Type u_1} {m0 : measurable_space α} (s : set α) :

Restrict a measure `μ` to a set `s` as an `ℝ≥0∞`-linear map.

Equations
noncomputable def measure_theory.measure.restrict {α : Type u_1} {m0 : measurable_space α} (μ : measure_theory.measure α) (s : set α) :

Restrict a measure `μ` to a set `s`.

Equations
Instances for `measure_theory.measure.restrict`
@[simp]
theorem measure_theory.measure.restrictₗ_apply {α : Type u_1} {m0 : measurable_space α} (s : set α) (μ : measure_theory.measure α) :

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_apply₀ {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (ht : (μ.restrict s)) :
(μ.restrict s) t = μ (t s)
@[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_apply₀' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (hs : μ) :
(μ.restrict s) t = μ (t s)
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
theorem measure_theory.measure.restrict_restrict₀ {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (hs : (μ.restrict t)) :
(μ.restrict t).restrict s = μ.restrict (s t)
@[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_restrict₀' {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (ht : μ) :
(μ.restrict t).restrict s = μ.restrict (s t)
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
@[simp]
@[simp]
theorem measure_theory.measure.restrict_inter_add_diff₀ {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {t : set α} (s : set α) (ht : μ) :
μ.restrict (s t) + μ.restrict (s \ t) = μ.restrict s
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_inter₀ {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {t : set α} (s : set α) (ht : μ) :
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} {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_union₀ {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s t : set α} (h : t) (ht : μ) :
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 : 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 : 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 ) (hm : (i : ι), ) {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 : 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} {μ : measure_theory.measure α} {f : α β} (hf : measurable f) {s : set β} (hs : measurable_set s) :
= (μ.restrict (f ⁻¹' 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_to_measurable {α : Type u_1} {m0 : measurable_space α} {μ : measure_theory.measure α} {s : set α} (h : μ s ) :
= μ.restrict s
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 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
theorem measure_theory.measure.restrict_Inf_eq_Inf_restrict {α : Type u_1} {t : set α} {m0 : measurable_space α} {m : set } (hm : m.nonempty) (ht : measurable_set t) :
(has_Inf.Inf m).restrict t = has_Inf.Inf ((λ (μ : , μ.restrict t) '' m)

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 = ) (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 = ) (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 = ) (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} (a : α) :

The dirac measure.

Equations
Instances for `measure_theory.measure.dirac`
@[protected, instance]
Equations
theorem measure_theory.measure.le_dirac_apply {α : Type u_1} {s : set α} {a : α} :
s.indicator 1 a
@[simp]
theorem measure_theory.measure.dirac_apply' {α : Type u_1} {s : set α} (a : α) (hs : measurable_set s) :
= s.indicator 1 a
@[simp]
theorem measure_theory.measure.dirac_apply_of_mem {α : Type u_1} {s : set α} {a : α} (h : a s) :
@[simp]
theorem measure_theory.measure.dirac_apply {α : Type u_1} (a : α) (s : set α) :
= s.indicator 1 a
theorem measure_theory.measure.map_dirac {α : Type u_1} {β : Type u_2} {f : α β} (hf : measurable f) (a : α) :
@[simp]
theorem measure_theory.measure.restrict_singleton {α : Type u_1} (μ : measure_theory.measure α) (a : α) :
μ.restrict {a} =
noncomputable def measure_theory.measure.sum {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} (f : ι ) :

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 ) (s : set α) :
∑' (i : ι), (f i) s
@[simp]
theorem measure_theory.measure.sum_apply {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} (f : ι ) {s : set α} (hs : measurable_set s) :
= ∑' (i : ι), (f i) s
theorem measure_theory.measure.le_sum {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} (μ : ι ) (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 } {s : set α} :
= 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 } {s : set α} (hs : measurable_set 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 ) :
measure_theory.measure.sum (λ (n : ι), = measure_theory.measure.sum (λ (m : ι'), measure_theory.measure.sum (λ (n : ι), μ n m))
theorem measure_theory.measure.ae_sum_iff {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} [countable ι] {μ : ι } {p : α Prop} :
(∀ᵐ (x : α) , 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 } {p : α Prop} (h : measurable_set {x : α | p x}) :
(∀ᵐ (x : α) , 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 ) :
= finset.univ.sum (λ (i : ι), μ i)
@[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.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 ) :
= (i : ι), (μ i).ae
@[simp]
theorem measure_theory.measure.sum_bool {α : Type u_1} {m0 : measurable_space α} (f : bool ) :
@[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 ) {s : set α} (hs : measurable_set s) :
= measure_theory.measure.sum (λ (i : ι), (μ i).restrict s)
@[simp]
theorem measure_theory.measure.sum_of_empty {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} [is_empty ι] (μ : ι ) :
theorem measure_theory.measure.sum_add_sum_compl {α : Type u_1} {ι : Type u_5} {m0 : measurable_space α} (s : set ι) (μ : ι ) :
theorem measure_theory.measure.sum_congr {α : Type u_1} {m0 : measurable_space α} {μ ν : } (h : (n : ), μ n = ν n) :
theorem measure_theory.measure.sum_add_sum {α : Type u_1} {m0 : measurable_space α} (μ ν : ) :
= measure_theory.measure.sum (λ (n : ), μ n + ν n)
theorem measure_theory.measure.map_eq_sum {α : Type u_1} {β : Type u_2} {m0 : measurable_space α} [countable β] (μ : measure_theory.measure α) (f : α β) (hf : measurable f) :
= measure_theory.measure.sum (λ (b : β), μ (f ⁻¹' {b})

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

@[simp]