mathlib3 documentation

measure_theory.measure.outer_measure

Outer Measures #

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

An outer measure is a function μ : set α → ℝ≥0∞, from the powerset of a type to the extended nonnegative real numbers that satisfies the following conditions:

  1. μ ∅ = 0;
  2. μ is monotone;
  3. μ is countably subadditive. This means that the outer measure of a countable union is at most the sum of the outer measure on the individual sets.

Note that we do not need α to be measurable to define an outer measure.

The outer measures on a type α form a complete lattice.

Given an arbitrary function m : set α → ℝ≥0∞ that sends ∅ to 0 we can define an outer measure on α that on s is defined to be the infimum of ∑ᵢ, m (sᵢ) for all collections of sets sᵢ that cover s. This is the unique maximal outer measure that is at most the given function. We also define this for functions m defined on a subset of set α, by treating the function as having value ∞ outside its domain.

Given an outer measure m, the Carathéodory-measurable sets are the sets s such that for all sets t we have m t = m (t ∩ s) + m (t \ s). This forms a measurable space.

Main definitions and statements #

References #

Tags #

outer measure, Carathéodory-measurable, Carathéodory's criterion

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structure measure_theory.outer_measure (α : Type u_1) :
Type u_1

An outer measure is a countably subadditive monotone function that sends ∅ to 0.

Instances for measure_theory.outer_measure
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@[protected, instance]
def measure_theory.outer_measure.has_coe_to_fun {α : Type u_1} :
has_coe_to_fun (measure_theory.outer_measure α) (λ (_x : measure_theory.outer_measure α), set α → ennreal)
Equations
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@[simp]
theorem measure_theory.outer_measure.measure_of_eq_coe {α : Type u_1} (m : measure_theory.outer_measure α) :
m.measure_of = ⇑m
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@[simp]
theorem measure_theory.outer_measure.empty' {α : Type u_1} (m : measure_theory.outer_measure α) :
⇑m ∅ = 0
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theorem measure_theory.outer_measure.mono' {α : Type u_1} (m : measure_theory.outer_measure α) {s₁ s₂ : set α} (h : s₁ ⊆ s₂) :
⇑m s₁ ≤ ⇑m s₂
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theorem measure_theory.outer_measure.mono_null {α : Type u_1} (m : measure_theory.outer_measure α) {s t : set α} (h : s ⊆ t) (ht : ⇑m t = 0) :
⇑m s = 0
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theorem measure_theory.outer_measure.pos_of_subset_ne_zero {α : Type u_1} (m : measure_theory.outer_measure α) {a b : set α} (hs : a ⊆ b) (hnz : ⇑m a ≠ 0) :
0 < ⇑m b
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@[protected]
theorem measure_theory.outer_measure.Union {α : Type u_1} (m : measure_theory.outer_measure α) {β : Type u_2} [countable β] (s : β → set α) :
⇑m (⋃ (i : β), s i) ≤ ∑' (i : β), ⇑m (s i)
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theorem measure_theory.outer_measure.Union_null {α : Type u_1} {β : Type u_2} [countable β] (m : measure_theory.outer_measure α) {s : β → set α} (h : ∀ (i : β), ⇑m (s i) = 0) :
⇑m (⋃ (i : β), s i) = 0
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@[simp]
theorem measure_theory.outer_measure.Union_null_iff {α : Type u_1} {β : Type u_2} [countable β] (m : measure_theory.outer_measure α) {s : β → set α} :
⇑m (⋃ (i : β), s i) = 0 ↔ ∀ (i : β), ⇑m (s i) = 0
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@[simp]
theorem measure_theory.outer_measure.Union_null_iff' {α : Type u_1} (m : measure_theory.outer_measure α) {ι : Prop} {s : ι → set α} :
⇑m (⋃ (i : ι), s i) = 0 ↔ ∀ (i : ι), ⇑m (s i) = 0

A version of Union_null_iff for unions indexed by Props. TODO: in the long run it would be better to combine this with Union_null_iff by generalising to Sort.

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theorem measure_theory.outer_measure.bUnion_null_iff {α : Type u_1} {β : Type u_2} (m : measure_theory.outer_measure α) {s : set β} (hs : s.countable) {t : β → set α} :
⇑m (⋃ (i : β) (H : i ∈ s), t i) = 0 ↔ ∀ (i : β), i ∈ s → ⇑m (t i) = 0
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theorem measure_theory.outer_measure.sUnion_null_iff {α : Type u_1} (m : measure_theory.outer_measure α) {S : set (set α)} (hS : S.countable) :
⇑m (⋃₀ S) = 0 ↔ ∀ (s : set α), s ∈ S → ⇑m s = 0
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@[protected]
theorem measure_theory.outer_measure.Union_finset {α : Type u_1} {β : Type u_2} (m : measure_theory.outer_measure α) (s : β → set α) (t : finset β) :
⇑m (⋃ (i : β) (H : i ∈ t), s i) ≤ t.sum (λ (i : β), ⇑m (s i))
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@[protected]
theorem measure_theory.outer_measure.union {α : Type u_1} (m : measure_theory.outer_measure α) (s₁ s₂ : set α) :
⇑m (s₁ ∪ s₂) ≤ ⇑m s₁ + ⇑m s₂
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theorem measure_theory.outer_measure.null_of_locally_null {α : Type u_1} [topological_space α] [topological_space.second_countable_topology α] (m : measure_theory.outer_measure α) (s : set α) (hs : ∀ (x : α), x ∈ s → (∃ (u : set α) (H : u ∈ nhds_within x s), ⇑m u = 0)) :
⇑m 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.

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theorem measure_theory.outer_measure.exists_mem_forall_mem_nhds_within_pos {α : Type u_1} [topological_space α] [topological_space.second_countable_topology α] (m : measure_theory.outer_measure α) {s : set α} (hs : ⇑m s ≠ 0) :
∃ (x : α) (H : x ∈ s), ∀ (t : set α), t ∈ nhds_within x s → 0 < ⇑m t

If m s ≠ 0, then for some point x ∈ s and any t ∈ 𝓝[s] x we have 0 < m t.

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theorem measure_theory.outer_measure.Union_of_tendsto_zero {α : Type u_1} {ι : Type u_2} (m : measure_theory.outer_measure α) {s : ι → set α} (l : filter ι) [l.ne_bot] (h0 : filter.tendsto (λ (k : ι), ⇑m ((⋃ (n : ι), s n) \ s k)) l (nhds 0)) :
⇑m (⋃ (n : ι), s n) = ⨆ (n : ι), ⇑m (s n)

If s : ι → set α is a sequence of sets, S = ⋃ n, s n, and m (S \ s n) tends to zero along some nontrivial filter (usually at_top on ι = ℕ), then m S = ⨆ n, m (s n).

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theorem measure_theory.outer_measure.Union_nat_of_monotone_of_tsum_ne_top {α : Type u_1} (m : measure_theory.outer_measure α) {s : ℕ → set α} (h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)) (h0 : ∑' (k : ℕ), ⇑m (s (k + 1) \ s k) ≠ ⊤) :
⇑m (⋃ (n : ℕ), s n) = ⨆ (n : ℕ), ⇑m (s n)

If s : ℕ → set α is a monotone sequence of sets such that ∑' k, m (s (k + 1) \ s k) ≠ ∞, then m (⋃ n, s n) = ⨆ n, m (s n).

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theorem measure_theory.outer_measure.le_inter_add_diff {α : Type u_1} {m : measure_theory.outer_measure α} {t : set α} (s : set α) :
⇑m t ≤ ⇑m (t ∩ s) + ⇑m (t \ s)
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theorem measure_theory.outer_measure.diff_null {α : Type u_1} (m : measure_theory.outer_measure α) (s : set α) {t : set α} (ht : ⇑m t = 0) :
⇑m (s \ t) = ⇑m s
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theorem measure_theory.outer_measure.union_null {α : Type u_1} (m : measure_theory.outer_measure α) {s₁ s₂ : set α} (h₁ : ⇑m s₁ = 0) (h₂ : ⇑m s₂ = 0) :
⇑m (s₁ ∪ s₂) = 0
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theorem measure_theory.outer_measure.coe_fn_injective {α : Type u_1} :
function.injective (λ (μ : measure_theory.outer_measure α) (s : set α), ⇑μ s)
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@[ext]
theorem measure_theory.outer_measure.ext {α : Type u_1} {μ₁ μ₂ : measure_theory.outer_measure α} (h : ∀ (s : set α), ⇑μ₁ s = ⇑μ₂ s) :
μ₁ = μ₂
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theorem measure_theory.outer_measure.ext_nonempty {α : Type u_1} {μ₁ μ₂ : measure_theory.outer_measure α} (h : ∀ (s : set α), s.nonempty → ⇑μ₁ s = ⇑μ₂ s) :
μ₁ = μ₂

A version of measure_theory.outer_measure.ext that assumes μ₁ s = μ₂ s on all nonempty sets s, and gets μ₁ ∅ = μ₂ ∅ from measure_theory.outer_measure.empty'.

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@[protected, instance]
def measure_theory.outer_measure.has_zero {α : Type u_1} :
has_zero (measure_theory.outer_measure α)
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@[simp]
theorem measure_theory.outer_measure.coe_zero {α : Type u_1} :
⇑0 = 0
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@[protected, instance]
def measure_theory.outer_measure.inhabited {α : Type u_1} :
inhabited (measure_theory.outer_measure α)
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@[protected, instance]
noncomputable def measure_theory.outer_measure.has_add {α : Type u_1} :
has_add (measure_theory.outer_measure α)
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@[simp]
theorem measure_theory.outer_measure.coe_add {α : Type u_1} (m₁ m₂ : measure_theory.outer_measure α) :
⇑(m₁ + m₂) = ⇑m₁ + ⇑m₂
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theorem measure_theory.outer_measure.add_apply {α : Type u_1} (m₁ m₂ : measure_theory.outer_measure α) (s : set α) :
⇑(m₁ + m₂) s = ⇑m₁ s + ⇑m₂ s
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@[protected, instance]
def measure_theory.outer_measure.has_smul {α : Type u_1} {R : Type u_3} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] :
has_smul R (measure_theory.outer_measure α)
Equations
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@[simp]
theorem measure_theory.outer_measure.coe_smul {α : Type u_1} {R : Type u_3} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] (c : R) (m : measure_theory.outer_measure α) :
⇑(c • m) = c • ⇑m
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theorem measure_theory.outer_measure.smul_apply {α : Type u_1} {R : Type u_3} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] (c : R) (m : measure_theory.outer_measure α) (s : set α) :
⇑(c • m) s = c • ⇑m s
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@[protected, instance]
def measure_theory.outer_measure.smul_comm_class {α : Type u_1} {R : Type u_3} {R' : Type u_4} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] [has_smul R' ennreal] [is_scalar_tower R' ennreal ennreal] [smul_comm_class R R' ennreal] :
smul_comm_class R R' (measure_theory.outer_measure α)
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@[protected, instance]
def measure_theory.outer_measure.is_scalar_tower {α : Type u_1} {R : Type u_3} {R' : Type u_4} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] [has_smul R' ennreal] [is_scalar_tower R' ennreal ennreal] [has_smul R R'] [is_scalar_tower R R' ennreal] :
is_scalar_tower R R' (measure_theory.outer_measure α)
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@[protected, instance]
def measure_theory.outer_measure.is_central_scalar {α : Type u_1} {R : Type u_3} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] [has_smul Rᵐᵒᵖ ennreal] [is_central_scalar R ennreal] :
is_central_scalar R (measure_theory.outer_measure α)
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@[protected, instance]
def measure_theory.outer_measure.mul_action {α : Type u_1} {R : Type u_3} [monoid R] [mul_action R ennreal] [is_scalar_tower R ennreal ennreal] :
mul_action R (measure_theory.outer_measure α)
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@[protected, instance]
noncomputable def measure_theory.outer_measure.add_comm_monoid {α : Type u_1} :
add_comm_monoid (measure_theory.outer_measure α)
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@[simp]
theorem measure_theory.outer_measure.coe_fn_add_monoid_hom_apply {α : Type u_1} (x : measure_theory.outer_measure α) (ᾰ : set α) :
⇑measure_theory.outer_measure.coe_fn_add_monoid_hom x ᾰ = ⇑x ᾰ
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def measure_theory.outer_measure.coe_fn_add_monoid_hom {α : Type u_1} :
measure_theory.outer_measure α →+ set α → ennreal

coe_fn as an add_monoid_hom.

Equations
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@[protected, instance]
noncomputable def measure_theory.outer_measure.distrib_mul_action {α : Type u_1} {R : Type u_3} [monoid R] [distrib_mul_action R ennreal] [is_scalar_tower R ennreal ennreal] :
distrib_mul_action R (measure_theory.outer_measure α)
Equations
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@[protected, instance]
noncomputable def measure_theory.outer_measure.module {α : Type u_1} {R : Type u_3} [semiring R] [module R ennreal] [is_scalar_tower R ennreal ennreal] :
module R (measure_theory.outer_measure α)
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@[protected, instance]
def measure_theory.outer_measure.has_bot {α : Type u_1} :
has_bot (measure_theory.outer_measure α)
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@[simp]
theorem measure_theory.outer_measure.coe_bot {α : Type u_1} :
⊥ = 0
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@[protected, instance]
def measure_theory.outer_measure.outer_measure.partial_order {α : Type u_1} :
partial_order (measure_theory.outer_measure α)
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@[protected, instance]
def measure_theory.outer_measure.outer_measure.order_bot {α : Type u_1} :
order_bot (measure_theory.outer_measure α)
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theorem measure_theory.outer_measure.univ_eq_zero_iff {α : Type u_1} (m : measure_theory.outer_measure α) :
⇑m set.univ = 0 ↔ m = 0
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@[protected, instance]
noncomputable def measure_theory.outer_measure.has_Sup {α : Type u_1} :
has_Sup (measure_theory.outer_measure α)
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@[protected, instance]
noncomputable def measure_theory.outer_measure.complete_lattice {α : Type u_1} :
complete_lattice (measure_theory.outer_measure α)
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@[simp]
theorem measure_theory.outer_measure.Sup_apply {α : Type u_1} (ms : set (measure_theory.outer_measure α)) (s : set α) :
⇑(has_Sup.Sup ms) s = ⨆ (m : measure_theory.outer_measure α) (H : m ∈ ms), ⇑m s
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@[simp]
theorem measure_theory.outer_measure.supr_apply {α : Type u_1} {ι : Sort u_2} (f : ι → measure_theory.outer_measure α) (s : set α) :
(⇑⨆ (i : ι), f i) s = ⨆ (i : ι), ⇑(f i) s
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@[norm_cast]
theorem measure_theory.outer_measure.coe_supr {α : Type u_1} {ι : Sort u_2} (f : ι → measure_theory.outer_measure α) :
(⇑⨆ (i : ι), f i) = ⨆ (i : ι), ⇑(f i)
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@[simp]
theorem measure_theory.outer_measure.sup_apply {α : Type u_1} (m₁ m₂ : measure_theory.outer_measure α) (s : set α) :
⇑(m₁ ⊔ m₂) s = ⇑m₁ s ⊔ ⇑m₂ s
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theorem measure_theory.outer_measure.smul_supr {α : Type u_1} {R : Type u_3} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] {ι : Sort u_2} (f : ι → measure_theory.outer_measure α) (c : R) :
(c • ⨆ (i : ι), f i) = ⨆ (i : ι), c • f i
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theorem measure_theory.outer_measure.mono'' {α : Type u_1} {m₁ m₂ : measure_theory.outer_measure α} {s₁ s₂ : set α} (hm : m₁ ≤ m₂) (hs : s₁ ⊆ s₂) :
⇑m₁ s₁ ≤ ⇑m₂ s₂
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def measure_theory.outer_measure.map {α : Type u_1} {β : Type u_2} (f : α → β) :
measure_theory.outer_measure α →ₗ[ennreal] measure_theory.outer_measure β

The pushforward of m along f. The outer measure on s is defined to be m (f ⁻¹' s).

Equations
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@[simp]
theorem measure_theory.outer_measure.map_apply {α : Type u_1} {β : Type u_2} (f : α → β) (m : measure_theory.outer_measure α) (s : set β) :
⇑(⇑(measure_theory.outer_measure.map f) m) s = ⇑m (f ⁻¹' s)
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@[simp]
theorem measure_theory.outer_measure.map_id {α : Type u_1} (m : measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.map id) m = m
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@[simp]
theorem measure_theory.outer_measure.map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β) (g : β → γ) (m : measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.map g) (⇑(measure_theory.outer_measure.map f) m) = ⇑(measure_theory.outer_measure.map (g ∘ f)) m
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theorem measure_theory.outer_measure.map_mono {α : Type u_1} {β : Type u_2} (f : α → β) :
monotone ⇑(measure_theory.outer_measure.map f)
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@[simp]
theorem measure_theory.outer_measure.map_sup {α : Type u_1} {β : Type u_2} (f : α → β) (m m' : measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.map f) (m ⊔ m') = ⇑(measure_theory.outer_measure.map f) m ⊔ ⇑(measure_theory.outer_measure.map f) m'
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@[simp]
theorem measure_theory.outer_measure.map_supr {α : Type u_1} {β : Type u_2} {ι : Sort u_3} (f : α → β) (m : ι → measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.map f) (⨆ (i : ι), m i) = ⨆ (i : ι), ⇑(measure_theory.outer_measure.map f) (m i)
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@[protected, instance]
noncomputable def measure_theory.outer_measure.functor  :
functor measure_theory.outer_measure
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@[protected, instance]
def measure_theory.outer_measure.is_lawful_functor  :
is_lawful_functor measure_theory.outer_measure
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noncomputable def measure_theory.outer_measure.dirac {α : Type u_1} (a : α) :
measure_theory.outer_measure α

The dirac outer measure.

Equations
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@[simp]
theorem measure_theory.outer_measure.dirac_apply {α : Type u_1} (a : α) (s : set α) :
⇑(measure_theory.outer_measure.dirac a) s = s.indicator (λ (_x : α), 1) a
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noncomputable def measure_theory.outer_measure.sum {α : Type u_1} {ι : Type u_2} (f : ι → measure_theory.outer_measure α) :
measure_theory.outer_measure α

The sum of an (arbitrary) collection of outer measures.

Equations
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@[simp]
theorem measure_theory.outer_measure.sum_apply {α : Type u_1} {ι : Type u_2} (f : ι → measure_theory.outer_measure α) (s : set α) :
⇑(measure_theory.outer_measure.sum f) s = ∑' (i : ι), ⇑(f i) s
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theorem measure_theory.outer_measure.smul_dirac_apply {α : Type u_1} (a : ennreal) (b : α) (s : set α) :
⇑(a • measure_theory.outer_measure.dirac b) s = s.indicator (λ (_x : α), a) b
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def measure_theory.outer_measure.comap {α : Type u_1} {β : Type u_2} (f : α → β) :
measure_theory.outer_measure β →ₗ[ennreal] measure_theory.outer_measure α

Pullback of an outer_measure: comap f μ s = μ (f '' s).

Equations
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@[simp]
theorem measure_theory.outer_measure.comap_apply {α : Type u_1} {β : Type u_2} (f : α → β) (m : measure_theory.outer_measure β) (s : set α) :
⇑(⇑(measure_theory.outer_measure.comap f) m) s = ⇑m (f '' s)
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theorem measure_theory.outer_measure.comap_mono {α : Type u_1} {β : Type u_2} (f : α → β) :
monotone ⇑(measure_theory.outer_measure.comap f)
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@[simp]
theorem measure_theory.outer_measure.comap_supr {α : Type u_1} {β : Type u_2} {ι : Sort u_3} (f : α → β) (m : ι → measure_theory.outer_measure β) :
⇑(measure_theory.outer_measure.comap f) (⨆ (i : ι), m i) = ⨆ (i : ι), ⇑(measure_theory.outer_measure.comap f) (m i)
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noncomputable def measure_theory.outer_measure.restrict {α : Type u_1} (s : set α) :
measure_theory.outer_measure α →ₗ[ennreal] measure_theory.outer_measure α

Restrict an outer_measure to a set.

Equations
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@[simp]
theorem measure_theory.outer_measure.restrict_apply {α : Type u_1} (s t : set α) (m : measure_theory.outer_measure α) :
⇑(⇑(measure_theory.outer_measure.restrict s) m) t = ⇑m (t ∩ s)
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theorem measure_theory.outer_measure.restrict_mono {α : Type u_1} {s t : set α} (h : s ⊆ t) {m m' : measure_theory.outer_measure α} (hm : m ≤ m') :
⇑(measure_theory.outer_measure.restrict s) m ≤ ⇑(measure_theory.outer_measure.restrict t) m'
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@[simp]
theorem measure_theory.outer_measure.restrict_univ {α : Type u_1} (m : measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.restrict set.univ) m = m
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@[simp]
theorem measure_theory.outer_measure.restrict_empty {α : Type u_1} (m : measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.restrict ∅) m = 0
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@[simp]
theorem measure_theory.outer_measure.restrict_supr {α : Type u_1} {ι : Sort u_2} (s : set α) (m : ι → measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.restrict s) (⨆ (i : ι), m i) = ⨆ (i : ι), ⇑(measure_theory.outer_measure.restrict s) (m i)
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theorem measure_theory.outer_measure.map_comap {α : Type u_1} {β : Type u_2} (f : α → β) (m : measure_theory.outer_measure β) :
⇑(measure_theory.outer_measure.map f) (⇑(measure_theory.outer_measure.comap f) m) = ⇑(measure_theory.outer_measure.restrict (set.range f)) m
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theorem measure_theory.outer_measure.map_comap_le {α : Type u_1} {β : Type u_2} (f : α → β) (m : measure_theory.outer_measure β) :
⇑(measure_theory.outer_measure.map f) (⇑(measure_theory.outer_measure.comap f) m) ≤ m
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theorem measure_theory.outer_measure.restrict_le_self {α : Type u_1} (m : measure_theory.outer_measure α) (s : set α) :
⇑(measure_theory.outer_measure.restrict s) m ≤ m
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@[simp]
theorem measure_theory.outer_measure.map_le_restrict_range {α : Type u_1} {β : Type u_2} {ma : measure_theory.outer_measure α} {mb : measure_theory.outer_measure β} {f : α → β} :
⇑(measure_theory.outer_measure.map f) ma ≤ ⇑(measure_theory.outer_measure.restrict (set.range f)) mb ↔ ⇑(measure_theory.outer_measure.map f) ma ≤ mb
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theorem measure_theory.outer_measure.map_comap_of_surjective {α : Type u_1} {β : Type u_2} {f : α → β} (hf : function.surjective f) (m : measure_theory.outer_measure β) :
⇑(measure_theory.outer_measure.map f) (⇑(measure_theory.outer_measure.comap f) m) = m
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theorem measure_theory.outer_measure.le_comap_map {α : Type u_1} {β : Type u_2} (f : α → β) (m : measure_theory.outer_measure α) :
m ≤ ⇑(measure_theory.outer_measure.comap f) (⇑(measure_theory.outer_measure.map f) m)
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theorem measure_theory.outer_measure.comap_map {α : Type u_1} {β : Type u_2} {f : α → β} (hf : function.injective f) (m : measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.comap f) (⇑(measure_theory.outer_measure.map f) m) = m
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@[simp]
theorem measure_theory.outer_measure.top_apply {α : Type u_1} {s : set α} (h : s.nonempty) :
⇑⊤ s = ⊤
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theorem measure_theory.outer_measure.top_apply' {α : Type u_1} (s : set α) :
⇑⊤ s = ⨅ (h : s = ∅), 0
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@[simp]
theorem measure_theory.outer_measure.comap_top {α : Type u_1} {β : Type u_2} (f : α → β) :
⇑(measure_theory.outer_measure.comap f) ⊤ = ⊤
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theorem measure_theory.outer_measure.map_top {α : Type u_1} {β : Type u_2} (f : α → β) :
⇑(measure_theory.outer_measure.map f) ⊤ = ⇑(measure_theory.outer_measure.restrict (set.range f)) ⊤
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theorem measure_theory.outer_measure.map_top_of_surjective {α : Type u_1} {β : Type u_2} (f : α → β) (hf : function.surjective f) :
⇑(measure_theory.outer_measure.map f) ⊤ = ⊤
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@[protected]
noncomputable def measure_theory.outer_measure.of_function {α : Type u_1} (m : set α → ennreal) (m_empty : m ∅ = 0) :
measure_theory.outer_measure α

Given any function m assigning measures to sets satisying m ∅ = 0, there is a unique maximal outer measure μ satisfying μ s ≤ m s for all s : set α.

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theorem measure_theory.outer_measure.of_function_apply {α : Type u_1} (m : set α → ennreal) (m_empty : m ∅ = 0) (s : set α) :
⇑(measure_theory.outer_measure.of_function m m_empty) s = ⨅ (t : ℕ → set α) (h : s ⊆ set.Union t), ∑' (n : ℕ), m (t n)
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theorem measure_theory.outer_measure.of_function_le {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} (s : set α) :
⇑(measure_theory.outer_measure.of_function m m_empty) s ≤ m s
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theorem measure_theory.outer_measure.of_function_eq {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} (s : set α) (m_mono : ∀ ⦃t : set α⦄, s ⊆ t → m s ≤ m t) (m_subadd : ∀ (s : ℕ → set α), m (⋃ (i : ℕ), s i) ≤ ∑' (i : ℕ), m (s i)) :
⇑(measure_theory.outer_measure.of_function m m_empty) s = m s
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theorem measure_theory.outer_measure.le_of_function {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} {μ : measure_theory.outer_measure α} :
μ ≤ measure_theory.outer_measure.of_function m m_empty ↔ ∀ (s : set α), ⇑μ s ≤ m s
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theorem measure_theory.outer_measure.is_greatest_of_function {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} :
is_greatest {μ : measure_theory.outer_measure α | ∀ (s : set α), ⇑μ s ≤ m s} (measure_theory.outer_measure.of_function m m_empty)
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theorem measure_theory.outer_measure.of_function_eq_Sup {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} :
measure_theory.outer_measure.of_function m m_empty = has_Sup.Sup {μ : measure_theory.outer_measure α | ∀ (s : set α), ⇑μ s ≤ m s}
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theorem measure_theory.outer_measure.of_function_union_of_top_of_nonempty_inter {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} {s t : set α} (h : ∀ (u : set α), (s ∩ u).nonempty → (t ∩ u).nonempty → m u = ⊤) :
⇑(measure_theory.outer_measure.of_function m m_empty) (s ∪ t) = ⇑(measure_theory.outer_measure.of_function m m_empty) s + ⇑(measure_theory.outer_measure.of_function m m_empty) t

If m u = ∞ for any set u that has nonempty intersection both with s and t, then μ (s ∪ t) = μ s + μ t, where μ = measure_theory.outer_measure.of_function m m_empty.

E.g., if α is an (e)metric space and m u = ∞ on any set of diameter ≥ r, then this lemma implies that μ (s ∪ t) = μ s + μ t on any two sets such that r ≤ edist x y for all x ∈ s and y ∈ t.

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theorem measure_theory.outer_measure.comap_of_function {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} {β : Type u_2} (f : β → α) (h : monotone m ∨ function.surjective f) :
⇑(measure_theory.outer_measure.comap f) (measure_theory.outer_measure.of_function m m_empty) = measure_theory.outer_measure.of_function (λ (s : set β), m (f '' s)) _
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theorem measure_theory.outer_measure.map_of_function_le {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} {β : Type u_2} (f : α → β) :
⇑(measure_theory.outer_measure.map f) (measure_theory.outer_measure.of_function m m_empty) ≤ measure_theory.outer_measure.of_function (λ (s : set β), m (f ⁻¹' s)) m_empty
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theorem measure_theory.outer_measure.map_of_function {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} {β : Type u_2} {f : α → β} (hf : function.injective f) :
⇑(measure_theory.outer_measure.map f) (measure_theory.outer_measure.of_function m m_empty) = measure_theory.outer_measure.of_function (λ (s : set β), m (f ⁻¹' s)) m_empty
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theorem measure_theory.outer_measure.restrict_of_function {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} (s : set α) (hm : monotone m) :
⇑(measure_theory.outer_measure.restrict s) (measure_theory.outer_measure.of_function m m_empty) = measure_theory.outer_measure.of_function (λ (t : set α), m (t ∩ s)) _
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theorem measure_theory.outer_measure.smul_of_function {α : Type u_1} {m : set α → ennreal} {m_empty : m ∅ = 0} {c : ennreal} (hc : c ≠ ⊤) :
c • measure_theory.outer_measure.of_function m m_empty = measure_theory.outer_measure.of_function (c • m) _
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noncomputable def measure_theory.outer_measure.bounded_by {α : Type u_1} (m : set α → ennreal) :
measure_theory.outer_measure α

Given any function m assigning measures to sets, there is a unique maximal outer measure μ satisfying μ s ≤ m s for all s : set α. This is the same as outer_measure.of_function, except that it doesn't require m ∅ = 0.

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theorem measure_theory.outer_measure.bounded_by_le {α : Type u_1} {m : set α → ennreal} (s : set α) :
⇑(measure_theory.outer_measure.bounded_by m) s ≤ m s
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theorem measure_theory.outer_measure.bounded_by_eq_of_function {α : Type u_1} {m : set α → ennreal} (m_empty : m ∅ = 0) (s : set α) :
⇑(measure_theory.outer_measure.bounded_by m) s = ⇑(measure_theory.outer_measure.of_function m m_empty) s
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theorem measure_theory.outer_measure.bounded_by_apply {α : Type u_1} {m : set α → ennreal} (s : set α) :
⇑(measure_theory.outer_measure.bounded_by m) s = ⨅ (t : ℕ → set α) (h : s ⊆ set.Union t), ∑' (n : ℕ), ⨆ (h : (t n).nonempty), m (t n)
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theorem measure_theory.outer_measure.bounded_by_eq {α : Type u_1} {m : set α → ennreal} (s : set α) (m_empty : m ∅ = 0) (m_mono : ∀ ⦃t : set α⦄, s ⊆ t → m s ≤ m t) (m_subadd : ∀ (s : ℕ → set α), m (⋃ (i : ℕ), s i) ≤ ∑' (i : ℕ), m (s i)) :
⇑(measure_theory.outer_measure.bounded_by m) s = m s
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@[simp]
theorem measure_theory.outer_measure.bounded_by_eq_self {α : Type u_1} (m : measure_theory.outer_measure α) :
measure_theory.outer_measure.bounded_by ⇑m = m
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theorem measure_theory.outer_measure.le_bounded_by {α : Type u_1} {m : set α → ennreal} {μ : measure_theory.outer_measure α} :
μ ≤ measure_theory.outer_measure.bounded_by m ↔ ∀ (s : set α), ⇑μ s ≤ m s
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theorem measure_theory.outer_measure.le_bounded_by' {α : Type u_1} {m : set α → ennreal} {μ : measure_theory.outer_measure α} :
μ ≤ measure_theory.outer_measure.bounded_by m ↔ ∀ (s : set α), s.nonempty → ⇑μ s ≤ m s
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@[simp]
theorem measure_theory.outer_measure.bounded_by_top {α : Type u_1} :
measure_theory.outer_measure.bounded_by ⊤ = ⊤
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@[simp]
theorem measure_theory.outer_measure.bounded_by_zero {α : Type u_1} :
measure_theory.outer_measure.bounded_by 0 = 0
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theorem measure_theory.outer_measure.smul_bounded_by {α : Type u_1} {m : set α → ennreal} {c : ennreal} (hc : c ≠ ⊤) :
c • measure_theory.outer_measure.bounded_by m = measure_theory.outer_measure.bounded_by (c • m)
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theorem measure_theory.outer_measure.comap_bounded_by {α : Type u_1} {m : set α → ennreal} {β : Type u_2} (f : β → α) (h : monotone (λ (s : {s // s.nonempty}), m ↑s) ∨ function.surjective f) :
⇑(measure_theory.outer_measure.comap f) (measure_theory.outer_measure.bounded_by m) = measure_theory.outer_measure.bounded_by (λ (s : set β), m (f '' s))
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theorem measure_theory.outer_measure.bounded_by_union_of_top_of_nonempty_inter {α : Type u_1} {m : set α → ennreal} {s t : set α} (h : ∀ (u : set α), (s ∩ u).nonempty → (t ∩ u).nonempty → m u = ⊤) :
⇑(measure_theory.outer_measure.bounded_by m) (s ∪ t) = ⇑(measure_theory.outer_measure.bounded_by m) s + ⇑(measure_theory.outer_measure.bounded_by m) t

If m u = ∞ for any set u that has nonempty intersection both with s and t, then μ (s ∪ t) = μ s + μ t, where μ = measure_theory.outer_measure.bounded_by m.

E.g., if α is an (e)metric space and m u = ∞ on any set of diameter ≥ r, then this lemma implies that μ (s ∪ t) = μ s + μ t on any two sets such that r ≤ edist x y for all x ∈ s and y ∈ t.

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def measure_theory.outer_measure.is_caratheodory {α : Type u} (m : measure_theory.outer_measure α) (s : set α) :
Prop

A set s is Carathéodory-measurable for an outer measure m if for all sets t we have m t = m (t ∩ s) + m (t \ s).

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theorem measure_theory.outer_measure.is_caratheodory_iff_le' {α : Type u} (m : measure_theory.outer_measure α) {s : set α} :
m.is_caratheodory s ↔ ∀ (t : set α), ⇑m (t ∩ s) + ⇑m (t \ s) ≤ ⇑m t
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@[simp]
theorem measure_theory.outer_measure.is_caratheodory_empty {α : Type u} (m : measure_theory.outer_measure α) :
m.is_caratheodory ∅
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theorem measure_theory.outer_measure.is_caratheodory_compl {α : Type u} (m : measure_theory.outer_measure α) {s₁ : set α} :
m.is_caratheodory s₁ → m.is_caratheodory s₁ᶜ
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@[simp]
theorem measure_theory.outer_measure.is_caratheodory_compl_iff {α : Type u} (m : measure_theory.outer_measure α) {s : set α} :
m.is_caratheodory sᶜ ↔ m.is_caratheodory s
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theorem measure_theory.outer_measure.is_caratheodory_union {α : Type u} (m : measure_theory.outer_measure α) {s₁ s₂ : set α} (h₁ : m.is_caratheodory s₁) (h₂ : m.is_caratheodory s₂) :
m.is_caratheodory (s₁ ∪ s₂)
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theorem measure_theory.outer_measure.measure_inter_union {α : Type u} (m : measure_theory.outer_measure α) {s₁ s₂ : set α} (h : s₁ ∩ s₂ ⊆ ∅) (h₁ : m.is_caratheodory s₁) {t : set α} :
⇑m (t ∩ (s₁ ∪ s₂)) = ⇑m (t ∩ s₁) + ⇑m (t ∩ s₂)
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theorem measure_theory.outer_measure.is_caratheodory_Union_lt {α : Type u} (m : measure_theory.outer_measure α) {s : ℕ → set α} {n : ℕ} :
(∀ (i : ℕ), i < n → m.is_caratheodory (s i)) → m.is_caratheodory (⋃ (i : ℕ) (H : i < n), s i)
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theorem measure_theory.outer_measure.is_caratheodory_inter {α : Type u} (m : measure_theory.outer_measure α) {s₁ s₂ : set α} (h₁ : m.is_caratheodory s₁) (h₂ : m.is_caratheodory s₂) :
m.is_caratheodory (s₁ ∩ s₂)
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theorem measure_theory.outer_measure.is_caratheodory_sum {α : Type u} (m : measure_theory.outer_measure α) {s : ℕ → set α} (h : ∀ (i : ℕ), m.is_caratheodory (s i)) (hd : pairwise (disjoint on s)) {t : set α} {n : ℕ} :
(finset.range n).sum (λ (i : ℕ), ⇑m (t ∩ s i)) = ⇑m (t ∩ ⋃ (i : ℕ) (H : i < n), s i)
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theorem measure_theory.outer_measure.is_caratheodory_Union_nat {α : Type u} (m : measure_theory.outer_measure α) {s : ℕ → set α} (h : ∀ (i : ℕ), m.is_caratheodory (s i)) (hd : pairwise (disjoint on s)) :
m.is_caratheodory (⋃ (i : ℕ), s i)
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theorem measure_theory.outer_measure.f_Union {α : Type u} (m : measure_theory.outer_measure α) {s : ℕ → set α} (h : ∀ (i : ℕ), m.is_caratheodory (s i)) (hd : pairwise (disjoint on s)) :
⇑m (⋃ (i : ℕ), s i) = ∑' (i : ℕ), ⇑m (s i)
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def measure_theory.outer_measure.caratheodory_dynkin {α : Type u} (m : measure_theory.outer_measure α) :
measurable_space.dynkin_system α

The Carathéodory-measurable sets for an outer measure m form a Dynkin system.

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@[protected]
def measure_theory.outer_measure.caratheodory {α : Type u} (m : measure_theory.outer_measure α) :
measurable_space α

Given an outer measure μ, the Carathéodory-measurable space is defined such that s is measurable if ∀t, μ t = μ (t ∩ s) + μ (t \ s).

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theorem measure_theory.outer_measure.is_caratheodory_iff {α : Type u} (m : measure_theory.outer_measure α) {s : set α} :
measurable_set s ↔ ∀ (t : set α), ⇑m t = ⇑m (t ∩ s) + ⇑m (t \ s)
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theorem measure_theory.outer_measure.is_caratheodory_iff_le {α : Type u} (m : measure_theory.outer_measure α) {s : set α} :
measurable_set s ↔ ∀ (t : set α), ⇑m (t ∩ s) + ⇑m (t \ s) ≤ ⇑m t
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@[protected]
theorem measure_theory.outer_measure.Union_eq_of_caratheodory {α : Type u} (m : measure_theory.outer_measure α) {s : ℕ → set α} (h : ∀ (i : ℕ), measurable_set (s i)) (hd : pairwise (disjoint on s)) :
⇑m (⋃ (i : ℕ), s i) = ∑' (i : ℕ), ⇑m (s i)
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theorem measure_theory.outer_measure.of_function_caratheodory {α : Type u_1} {m : set α → ennreal} {s : set α} {h₀ : m ∅ = 0} (hs : ∀ (t : set α), m (t ∩ s) + m (t \ s) ≤ m t) :
measurable_set s
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theorem measure_theory.outer_measure.bounded_by_caratheodory {α : Type u_1} {m : set α → ennreal} {s : set α} (hs : ∀ (t : set α), m (t ∩ s) + m (t \ s) ≤ m t) :
measurable_set s
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@[simp]
theorem measure_theory.outer_measure.zero_caratheodory {α : Type u_1} :
0.caratheodory = ⊤
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theorem measure_theory.outer_measure.top_caratheodory {α : Type u_1} :
⊤.caratheodory = ⊤
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theorem measure_theory.outer_measure.le_add_caratheodory {α : Type u_1} (m₁ m₂ : measure_theory.outer_measure α) :
m₁.caratheodory ⊓ m₂.caratheodory ≤ (m₁ + m₂).caratheodory
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theorem measure_theory.outer_measure.le_sum_caratheodory {α : Type u_1} {ι : Type u_2} (m : ι → measure_theory.outer_measure α) :
(⨅ (i : ι), (m i).caratheodory) ≤ (measure_theory.outer_measure.sum m).caratheodory
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theorem measure_theory.outer_measure.le_smul_caratheodory {α : Type u_1} (a : ennreal) (m : measure_theory.outer_measure α) :
m.caratheodory ≤ (a • m).caratheodory
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@[simp]
theorem measure_theory.outer_measure.dirac_caratheodory {α : Type u_1} (a : α) :
(measure_theory.outer_measure.dirac a).caratheodory = ⊤
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noncomputable def measure_theory.outer_measure.Inf_gen {α : Type u_1} (m : set (measure_theory.outer_measure α)) (s : set α) :
ennreal

Given a set of outer measures, we define a new function that on a set s is defined to be the infimum of μ(s) for the outer measures μ in the collection. We ensure that this function is defined to be 0 on ∅, even if the collection of outer measures is empty. The outer measure generated by this function is the infimum of the given outer measures.

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theorem measure_theory.outer_measure.Inf_gen_def {α : Type u_1} (m : set (measure_theory.outer_measure α)) (t : set α) :
measure_theory.outer_measure.Inf_gen m t = ⨅ (μ : measure_theory.outer_measure α) (h : μ ∈ m), ⇑μ t
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theorem measure_theory.outer_measure.Inf_eq_bounded_by_Inf_gen {α : Type u_1} (m : set (measure_theory.outer_measure α)) :
has_Inf.Inf m = measure_theory.outer_measure.bounded_by (measure_theory.outer_measure.Inf_gen m)
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theorem measure_theory.outer_measure.supr_Inf_gen_nonempty {α : Type u_1} {m : set (measure_theory.outer_measure α)} (h : m.nonempty) (t : set α) :
(⨆ (h : t.nonempty), measure_theory.outer_measure.Inf_gen m t) = ⨅ (μ : measure_theory.outer_measure α) (h : μ ∈ m), ⇑μ t
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theorem measure_theory.outer_measure.Inf_apply {α : Type u_1} {m : set (measure_theory.outer_measure α)} {s : set α} (h : m.nonempty) :
⇑(has_Inf.Inf m) s = ⨅ (t : ℕ → set α) (h2 : s ⊆ set.Union t), ∑' (n : ℕ), ⨅ (μ : measure_theory.outer_measure α) (h3 : μ ∈ m), ⇑μ (t n)

The value of the Infimum of a nonempty set of outer measures on a set is not simply the minimum value of a measure on that set: it is the infimum sum of measures of countable set of sets that covers that set, where a different measure can be used for each set in the cover.

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theorem measure_theory.outer_measure.Inf_apply' {α : Type u_1} {m : set (measure_theory.outer_measure α)} {s : set α} (h : s.nonempty) :
⇑(has_Inf.Inf m) s = ⨅ (t : ℕ → set α) (h2 : s ⊆ set.Union t), ∑' (n : ℕ), ⨅ (μ : measure_theory.outer_measure α) (h3 : μ ∈ m), ⇑μ (t n)

The value of the Infimum of a set of outer measures on a nonempty set is not simply the minimum value of a measure on that set: it is the infimum sum of measures of countable set of sets that covers that set, where a different measure can be used for each set in the cover.

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theorem measure_theory.outer_measure.infi_apply {α : Type u_1} {ι : Sort u_2} [nonempty ι] (m : ι → measure_theory.outer_measure α) (s : set α) :
(⇑⨅ (i : ι), m i) s = ⨅ (t : ℕ → set α) (h2 : s ⊆ set.Union t), ∑' (n : ℕ), ⨅ (i : ι), ⇑(m i) (t n)

The value of the Infimum of a nonempty family of outer measures on a set is not simply the minimum value of a measure on that set: it is the infimum sum of measures of countable set of sets that covers that set, where a different measure can be used for each set in the cover.

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theorem measure_theory.outer_measure.infi_apply' {α : Type u_1} {ι : Sort u_2} (m : ι → measure_theory.outer_measure α) {s : set α} (hs : s.nonempty) :
(⇑⨅ (i : ι), m i) s = ⨅ (t : ℕ → set α) (h2 : s ⊆ set.Union t), ∑' (n : ℕ), ⨅ (i : ι), ⇑(m i) (t n)

The value of the Infimum of a family of outer measures on a nonempty set is not simply the minimum value of a measure on that set: it is the infimum sum of measures of countable set of sets that covers that set, where a different measure can be used for each set in the cover.

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theorem measure_theory.outer_measure.binfi_apply {α : Type u_1} {ι : Type u_2} {I : set ι} (hI : I.nonempty) (m : ι → measure_theory.outer_measure α) (s : set α) :
(⇑⨅ (i : ι) (H : i ∈ I), m i) s = ⨅ (t : ℕ → set α) (h2 : s ⊆ set.Union t), ∑' (n : ℕ), ⨅ (i : ι) (H : i ∈ I), ⇑(m i) (t n)

The value of the Infimum of a nonempty family of outer measures on a set is not simply the minimum value of a measure on that set: it is the infimum sum of measures of countable set of sets that covers that set, where a different measure can be used for each set in the cover.

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theorem measure_theory.outer_measure.binfi_apply' {α : Type u_1} {ι : Type u_2} (I : set ι) (m : ι → measure_theory.outer_measure α) {s : set α} (hs : s.nonempty) :
(⇑⨅ (i : ι) (H : i ∈ I), m i) s = ⨅ (t : ℕ → set α) (h2 : s ⊆ set.Union t), ∑' (n : ℕ), ⨅ (i : ι) (H : i ∈ I), ⇑(m i) (t n)

The value of the Infimum of a nonempty family of outer measures on a set is not simply the minimum value of a measure on that set: it is the infimum sum of measures of countable set of sets that covers that set, where a different measure can be used for each set in the cover.

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theorem measure_theory.outer_measure.map_infi_le {α : Type u_1} {ι : Sort u_2} {β : Type u_3} (f : α → β) (m : ι → measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.map f) (⨅ (i : ι), m i) ≤ ⨅ (i : ι), ⇑(measure_theory.outer_measure.map f) (m i)
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theorem measure_theory.outer_measure.comap_infi {α : Type u_1} {ι : Sort u_2} {β : Type u_3} (f : α → β) (m : ι → measure_theory.outer_measure β) :
⇑(measure_theory.outer_measure.comap f) (⨅ (i : ι), m i) = ⨅ (i : ι), ⇑(measure_theory.outer_measure.comap f) (m i)
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theorem measure_theory.outer_measure.map_infi {α : Type u_1} {ι : Sort u_2} {β : Type u_3} {f : α → β} (hf : function.injective f) (m : ι → measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.map f) (⨅ (i : ι), m i) = ⇑(measure_theory.outer_measure.restrict (set.range f)) (⨅ (i : ι), ⇑(measure_theory.outer_measure.map f) (m i))
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theorem measure_theory.outer_measure.map_infi_comap {α : Type u_1} {ι : Sort u_2} {β : Type u_3} [nonempty ι] {f : α → β} (m : ι → measure_theory.outer_measure β) :
⇑(measure_theory.outer_measure.map f) (⨅ (i : ι), ⇑(measure_theory.outer_measure.comap f) (m i)) = ⨅ (i : ι), ⇑(measure_theory.outer_measure.map f) (⇑(measure_theory.outer_measure.comap f) (m i))
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theorem measure_theory.outer_measure.map_binfi_comap {α : Type u_1} {ι : Type u_2} {β : Type u_3} {I : set ι} (hI : I.nonempty) {f : α → β} (m : ι → measure_theory.outer_measure β) :
⇑(measure_theory.outer_measure.map f) (⨅ (i : ι) (H : i ∈ I), ⇑(measure_theory.outer_measure.comap f) (m i)) = ⨅ (i : ι) (H : i ∈ I), ⇑(measure_theory.outer_measure.map f) (⇑(measure_theory.outer_measure.comap f) (m i))
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theorem measure_theory.outer_measure.restrict_infi_restrict {α : Type u_1} {ι : Sort u_2} (s : set α) (m : ι → measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.restrict s) (⨅ (i : ι), ⇑(measure_theory.outer_measure.restrict s) (m i)) = ⇑(measure_theory.outer_measure.restrict s) (⨅ (i : ι), m i)
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theorem measure_theory.outer_measure.restrict_infi {α : Type u_1} {ι : Sort u_2} [nonempty ι] (s : set α) (m : ι → measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.restrict s) (⨅ (i : ι), m i) = ⨅ (i : ι), ⇑(measure_theory.outer_measure.restrict s) (m i)
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theorem measure_theory.outer_measure.restrict_binfi {α : Type u_1} {ι : Type u_2} {I : set ι} (hI : I.nonempty) (s : set α) (m : ι → measure_theory.outer_measure α) :
⇑(measure_theory.outer_measure.restrict s) (⨅ (i : ι) (H : i ∈ I), m i) = ⨅ (i : ι) (H : i ∈ I), ⇑(measure_theory.outer_measure.restrict s) (m i)
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theorem measure_theory.outer_measure.restrict_Inf_eq_Inf_restrict {α : Type u_1} (m : set (measure_theory.outer_measure α)) {s : set α} (hm : m.nonempty) :
⇑(measure_theory.outer_measure.restrict s) (has_Inf.Inf m) = has_Inf.Inf (⇑(measure_theory.outer_measure.restrict s) '' m)

This proves that Inf and restrict commute for outer measures, so long as the set of outer measures is nonempty.

Induced Outer Measure #

We can extend a function defined on a subset of set α to an outer measure. The underlying function is called extend, and the measure it induces is called induced_outer_measure.

Some lemmas below are proven twice, once in the general case, and one where the function m is only defined on measurable sets (i.e. when P = measurable_set). In the latter cases, we can remove some hypotheses in the statement. The general version has the same name, but with a prime at the end.

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noncomputable def measure_theory.extend {α : Type u_1} {P : α → Prop} (m : Π (s : α), P s → ennreal) (s : α) :
ennreal

We can trivially extend a function defined on a subclass of objects (with codomain ℝ≥0∞) to all objects by defining it to be ∞ on the objects not in the class.

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theorem measure_theory.smul_extend {α : Type u_1} {P : α → Prop} (m : Π (s : α), P s → ennreal) {R : Type u_2} [has_zero R] [smul_with_zero R ennreal] [is_scalar_tower R ennreal ennreal] [no_zero_smul_divisors R ennreal] {c : R} (hc : c ≠ 0) :
c • measure_theory.extend m = measure_theory.extend (λ (s : α) (h : P s), c • m s h)
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theorem measure_theory.extend_eq {α : Type u_1} {P : α → Prop} (m : Π (s : α), P s → ennreal) {s : α} (h : P s) :
measure_theory.extend m s = m s h
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theorem measure_theory.extend_eq_top {α : Type u_1} {P : α → Prop} (m : Π (s : α), P s → ennreal) {s : α} (h : ¬P s) :
measure_theory.extend m s = ⊤
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theorem measure_theory.le_extend {α : Type u_1} {P : α → Prop} (m : Π (s : α), P s → ennreal) {s : α} (h : P s) :
m s h ≤ measure_theory.extend m s
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theorem measure_theory.extend_congr {α : Type u_1} {P : α → Prop} (m : Π (s : α), P s → ennreal) {β : Type u_2} {Pb : β → Prop} {mb : Π (s : β), Pb s → ennreal} {sa : α} {sb : β} (hP : P sa ↔ Pb sb) (hm : ∀ (ha : P sa) (hb : Pb sb), m sa ha = mb sb hb) :
measure_theory.extend m sa = measure_theory.extend mb sb
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@[simp]
theorem measure_theory.extend_top {α : Type u_1} {P : α → Prop} :
measure_theory.extend (λ (s : α) (h : P s), ⊤) = ⊤
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theorem measure_theory.extend_empty {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} (P0 : P ∅) (m0 : m ∅ P0 = 0) :
measure_theory.extend m ∅ = 0
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theorem measure_theory.extend_Union_nat {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) {f : ℕ → set α} (hm : ∀ (i : ℕ), P (f i)) (mU : m (⋃ (i : ℕ), f i) _ = ∑' (i : ℕ), m (f i) _) :
measure_theory.extend m (⋃ (i : ℕ), f i) = ∑' (i : ℕ), measure_theory.extend m (f i)
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theorem measure_theory.extend_Union_le_tsum_nat' {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) (msU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ (i : ℕ), f i) _ ≤ ∑' (i : ℕ), m (f i) _) (s : ℕ → set α) :
measure_theory.extend m (⋃ (i : ℕ), s i) ≤ ∑' (i : ℕ), measure_theory.extend m (s i)
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theorem measure_theory.extend_mono' {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} (m_mono : ∀ ⦃s₁ s₂ : set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) ⦃s₁ s₂ : set α⦄ (h₁ : P s₁) (hs : s₁ ⊆ s₂) :
measure_theory.extend m s₁ ≤ measure_theory.extend m s₂
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theorem measure_theory.extend_Union {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} (P0 : P ∅) (m0 : m ∅ P0 = 0) (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) (mU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), P (f i)), pairwise (disjoint on f) → m (⋃ (i : ℕ), f i) _ = ∑' (i : ℕ), m (f i) _) {β : Type u_2} [countable β] {f : β → set α} (hd : pairwise (disjoint on f)) (hm : ∀ (i : β), P (f i)) :
measure_theory.extend m (⋃ (i : β), f i) = ∑' (i : β), measure_theory.extend m (f i)
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theorem measure_theory.extend_union {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} (P0 : P ∅) (m0 : m ∅ P0 = 0) (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) (mU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), P (f i)), pairwise (disjoint on f) → m (⋃ (i : ℕ), f i) _ = ∑' (i : ℕ), m (f i) _) {s₁ s₂ : set α} (hd : disjoint s₁ s₂) (h₁ : P s₁) (h₂ : P s₂) :
measure_theory.extend m (s₁ ∪ s₂) = measure_theory.extend m s₁ + measure_theory.extend m s₂
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noncomputable def measure_theory.induced_outer_measure {α : Type u_1} {P : set α → Prop} (m : Π (s : set α), P s → ennreal) (P0 : P ∅) (m0 : m ∅ P0 = 0) :
measure_theory.outer_measure α

Given an arbitrary function on a subset of sets, we can define the outer measure corresponding to it (this is the unique maximal outer measure that is at most m on the domain of m).

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theorem measure_theory.le_induced_outer_measure {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} {P0 : P ∅} {m0 : m ∅ P0 = 0} {μ : measure_theory.outer_measure α} :
μ ≤ measure_theory.induced_outer_measure m P0 m0 ↔ ∀ (s : set α) (hs : P s), ⇑μ s ≤ m s hs
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theorem measure_theory.induced_outer_measure_union_of_false_of_nonempty_inter {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} {P0 : P ∅} {m0 : m ∅ P0 = 0} {s t : set α} (h : ∀ (u : set α), (s ∩ u).nonempty → (t ∩ u).nonempty → ¬P u) :
⇑(measure_theory.induced_outer_measure m P0 m0) (s ∪ t) = ⇑(measure_theory.induced_outer_measure m P0 m0) s + ⇑(measure_theory.induced_outer_measure m P0 m0) t

If P u is false for any set u that has nonempty intersection both with s and t, then μ (s ∪ t) = μ s + μ t, where μ = induced_outer_measure m P0 m0.

E.g., if α is an (e)metric space and P u = diam u < r, then this lemma implies that μ (s ∪ t) = μ s + μ t on any two sets such that r ≤ edist x y for all x ∈ s and y ∈ t.

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theorem measure_theory.induced_outer_measure_eq_extend' {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) (msU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ (i : ℕ), f i) _ ≤ ∑' (i : ℕ), m (f i) _) (m_mono : ∀ ⦃s₁ s₂ : set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) {s : set α} (hs : P s) :
⇑(measure_theory.induced_outer_measure m P0 m0) s = measure_theory.extend m s
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theorem measure_theory.induced_outer_measure_eq' {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) (msU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ (i : ℕ), f i) _ ≤ ∑' (i : ℕ), m (f i) _) (m_mono : ∀ ⦃s₁ s₂ : set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) {s : set α} (hs : P s) :
⇑(measure_theory.induced_outer_measure m P0 m0) s = m s hs
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theorem measure_theory.induced_outer_measure_eq_infi {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) (msU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ (i : ℕ), f i) _ ≤ ∑' (i : ℕ), m (f i) _) (m_mono : ∀ ⦃s₁ s₂ : set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) (s : set α) :
⇑(measure_theory.induced_outer_measure m P0 m0) s = ⨅ (t : set α) (ht : P t) (h : s ⊆ t), m t ht
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theorem measure_theory.induced_outer_measure_preimage {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) (msU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ (i : ℕ), f i) _ ≤ ∑' (i : ℕ), m (f i) _) (m_mono : ∀ ⦃s₁ s₂ : set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) (f : α ≃ α) (Pm : ∀ (s : set α), P (⇑f ⁻¹' s) ↔ P s) (mm : ∀ (s : set α) (hs : P s), m (⇑f ⁻¹' s) _ = m s hs) {A : set α} :
⇑(measure_theory.induced_outer_measure m P0 m0) (⇑f ⁻¹' A) = ⇑(measure_theory.induced_outer_measure m P0 m0) A
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theorem measure_theory.induced_outer_measure_exists_set {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) (msU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ (i : ℕ), f i) _ ≤ ∑' (i : ℕ), m (f i) _) (m_mono : ∀ ⦃s₁ s₂ : set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) {s : set α} (hs : ⇑(measure_theory.induced_outer_measure m P0 m0) s ≠ ⊤) {ε : ennreal} (hε : ε ≠ 0) :
∃ (t : set α) (ht : P t), s ⊆ t ∧ ⇑(measure_theory.induced_outer_measure m P0 m0) t ≤ ⇑(measure_theory.induced_outer_measure m P0 m0) s + ε
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theorem measure_theory.induced_outer_measure_caratheodory {α : Type u_1} {P : set α → Prop} {m : Π (s : set α), P s → ennreal} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)) (msU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ (i : ℕ), f i) _ ≤ ∑' (i : ℕ), m (f i) _) (m_mono : ∀ ⦃s₁ s₂ : set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) (s : set α) :
measurable_set s ↔ ∀ (t : set α), P t → ⇑(measure_theory.induced_outer_measure m P0 m0) (t ∩ s) + ⇑(measure_theory.induced_outer_measure m P0 m0) (t \ s) ≤ ⇑(measure_theory.induced_outer_measure m P0 m0) t

To test whether s is Carathéodory-measurable we only need to check the sets t for which P t holds. See of_function_caratheodory for another way to show the Carathéodory-measurability of s.

If P is measurable_set for some measurable space, then we can remove some hypotheses of the above lemmas.

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theorem measure_theory.extend_mono {α : Type u_1} [measurable_space α] {m : Π (s : set α), measurable_set s → ennreal} (m0 : m ∅ measurable_set.empty = 0) (mU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), measurable_set (f i)), pairwise (disjoint on f) → m (⋃ (i : ℕ), f i) _ = ∑' (i : ℕ), m (f i) _) {s₁ s₂ : set α} (h₁ : measurable_set s₁) (hs : s₁ ⊆ s₂) :
measure_theory.extend m s₁ ≤ measure_theory.extend m s₂
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theorem measure_theory.extend_Union_le_tsum_nat {α : Type u_1} [measurable_space α] {m : Π (s : set α), measurable_set s → ennreal} (m0 : m ∅ measurable_set.empty = 0) (mU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), measurable_set (f i)), pairwise (disjoint on f) → m (⋃ (i : ℕ), f i) _ = ∑' (i : ℕ), m (f i) _) (s : ℕ → set α) :
measure_theory.extend m (⋃ (i : ℕ), s i) ≤ ∑' (i : ℕ), measure_theory.extend m (s i)
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theorem measure_theory.induced_outer_measure_eq_extend {α : Type u_1} [measurable_space α] {m : Π (s : set α), measurable_set s → ennreal} (m0 : m ∅ measurable_set.empty = 0) (mU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), measurable_set (f i)), pairwise (disjoint on f) → m (⋃ (i : ℕ), f i) _ = ∑' (i : ℕ), m (f i) _) {s : set α} (hs : measurable_set s) :
⇑(measure_theory.induced_outer_measure m measurable_set.empty m0) s = measure_theory.extend m s
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theorem measure_theory.induced_outer_measure_eq {α : Type u_1} [measurable_space α] {m : Π (s : set α), measurable_set s → ennreal} (m0 : m ∅ measurable_set.empty = 0) (mU : ∀ ⦃f : ℕ → set α⦄ (hm : ∀ (i : ℕ), measurable_set (f i)), pairwise (disjoint on f) → m (⋃ (i : ℕ), f i) _ = ∑' (i : ℕ), m (f i) _) {s : set α} (hs : measurable_set s) :
⇑(measure_theory.induced_outer_measure m measurable_set.empty m0) s = m s hs
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noncomputable def measure_theory.outer_measure.trim {α : Type u_1} [measurable_space α] (m : measure_theory.outer_measure α) :
measure_theory.outer_measure α

Given an outer measure m we can forget its value on non-measurable sets, and then consider m.trim, the unique maximal outer measure less than that function.

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theorem measure_theory.outer_measure.le_trim {α : Type u_1} [measurable_space α] (m : measure_theory.outer_measure α) :
m ≤ m.trim
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theorem measure_theory.outer_measure.trim_eq {α : Type u_1} [measurable_space α] (m : measure_theory.outer_measure α) {s : set α} (hs : measurable_set s) :
⇑(m.trim) s = ⇑m s
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theorem measure_theory.outer_measure.trim_congr {α : Type u_1} [measurable_space α] {m₁ m₂ : measure_theory.outer_measure α} (H : ∀ {s : set α}, measurable_set s → ⇑m₁ s = ⇑m₂ s) :
m₁.trim = m₂.trim
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theorem measure_theory.outer_measure.trim_mono {α : Type u_1} [measurable_space α] :
monotone measure_theory.outer_measure.trim
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theorem measure_theory.outer_measure.le_trim_iff {α : Type u_1} [measurable_space α] {m₁ m₂ : measure_theory.outer_measure α} :
m₁ ≤ m₂.trim ↔ ∀ (s : set α), measurable_set s → ⇑m₁ s ≤ ⇑m₂ s
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theorem measure_theory.outer_measure.trim_le_trim_iff {α : Type u_1} [measurable_space α] {m₁ m₂ : measure_theory.outer_measure α} :
m₁.trim ≤ m₂.trim ↔ ∀ (s : set α), measurable_set s → ⇑m₁ s ≤ ⇑m₂ s
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theorem measure_theory.outer_measure.trim_eq_trim_iff {α : Type u_1} [measurable_space α] {m₁ m₂ : measure_theory.outer_measure α} :
m₁.trim = m₂.trim ↔ ∀ (s : set α), measurable_set s → ⇑m₁ s = ⇑m₂ s
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theorem measure_theory.outer_measure.trim_eq_infi {α : Type u_1} [measurable_space α] (m : measure_theory.outer_measure α) (s : set α) :
⇑(m.trim) s = ⨅ (t : set α) (st : s ⊆ t) (ht : measurable_set t), ⇑m t
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theorem measure_theory.outer_measure.trim_eq_infi' {α : Type u_1} [measurable_space α] (m : measure_theory.outer_measure α) (s : set α) :
⇑(m.trim) s = ⨅ (t : {t // s ⊆ t ∧ measurable_set t}), ⇑m ↑t
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theorem measure_theory.outer_measure.trim_trim {α : Type u_1} [measurable_space α] (m : measure_theory.outer_measure α) :
m.trim.trim = m.trim
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@[simp]
theorem measure_theory.outer_measure.trim_top {α : Type u_1} [measurable_space α] :
⊤.trim = ⊤
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@[simp]
theorem measure_theory.outer_measure.trim_zero {α : Type u_1} [measurable_space α] :
0.trim = 0
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theorem measure_theory.outer_measure.trim_sum_ge {α : Type u_1} [measurable_space α] {ι : Type u_2} (m : ι → measure_theory.outer_measure α) :
measure_theory.outer_measure.sum (λ (i : ι), (m i).trim) ≤ (measure_theory.outer_measure.sum m).trim
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theorem measure_theory.outer_measure.exists_measurable_superset_eq_trim {α : Type u_1} [measurable_space α] (m : measure_theory.outer_measure α) (s : set α) :
∃ (t : set α), s ⊆ t ∧ measurable_set t ∧ ⇑m t = ⇑(m.trim) s
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theorem measure_theory.outer_measure.exists_measurable_superset_of_trim_eq_zero {α : Type u_1} [measurable_space α] {m : measure_theory.outer_measure α} {s : set α} (h : ⇑(m.trim) s = 0) :
∃ (t : set α), s ⊆ t ∧ measurable_set t ∧ ⇑m t = 0
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theorem measure_theory.outer_measure.exists_measurable_superset_forall_eq_trim {α : Type u_1} [measurable_space α] {ι : Sort u_2} [countable ι] (μ : ι → measure_theory.outer_measure α) (s : set α) :
∃ (t : set α), s ⊆ t ∧ measurable_set t ∧ ∀ (i : ι), ⇑(μ i) t = ⇑((μ i).trim) s

If μ i is a countable family of outer measures, then for every set s there exists a measurable set t ⊇ s such that μ i t = (μ i).trim s for all i.

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theorem measure_theory.outer_measure.trim_binop {α : Type u_1} [measurable_space α] {m₁ m₂ m₃ : measure_theory.outer_measure α} {op : ennreal → ennreal → ennreal} (h : ∀ (s : set α), ⇑m₁ s = op (⇑m₂ s) (⇑m₃ s)) (s : set α) :
⇑(m₁.trim) s = op (⇑(m₂.trim) s) (⇑(m₃.trim) s)

If m₁ s = op (m₂ s) (m₃ s) for all s, then the same is true for m₁.trim, m₂.trim, and m₃ s.

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theorem measure_theory.outer_measure.trim_op {α : Type u_1} [measurable_space α] {m₁ m₂ : measure_theory.outer_measure α} {op : ennreal → ennreal} (h : ∀ (s : set α), ⇑m₁ s = op (⇑m₂ s)) (s : set α) :
⇑(m₁.trim) s = op (⇑(m₂.trim) s)

If m₁ s = op (m₂ s) for all s, then the same is true for m₁.trim and m₂.trim.

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theorem measure_theory.outer_measure.trim_add {α : Type u_1} [measurable_space α] (m₁ m₂ : measure_theory.outer_measure α) :
(m₁ + m₂).trim = m₁.trim + m₂.trim

trim is additive.

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theorem measure_theory.outer_measure.trim_smul {α : Type u_1} [measurable_space α] {R : Type u_2} [has_smul R ennreal] [is_scalar_tower R ennreal ennreal] (c : R) (m : measure_theory.outer_measure α) :
(c • m).trim = c • m.trim

trim respects scalar multiplication.

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theorem measure_theory.outer_measure.trim_sup {α : Type u_1} [measurable_space α] (m₁ m₂ : measure_theory.outer_measure α) :
(m₁ ⊔ m₂).trim = m₁.trim ⊔ m₂.trim

trim sends the supremum of two outer measures to the supremum of the trimmed measures.

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theorem measure_theory.outer_measure.trim_supr {α : Type u_1} [measurable_space α] {ι : Sort u_2} [countable ι] (μ : ι → measure_theory.outer_measure α) :
(⨆ (i : ι), μ i).trim = ⨆ (i : ι), (μ i).trim

trim sends the supremum of a countable family of outer measures to the supremum of the trimmed measures.

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theorem measure_theory.outer_measure.restrict_trim {α : Type u_1} [measurable_space α] {μ : measure_theory.outer_measure α} {s : set α} (hs : measurable_set s) :
(⇑(measure_theory.outer_measure.restrict s) μ).trim = ⇑(measure_theory.outer_measure.restrict s) μ.trim

The trimmed property of a measure μ states that μ.to_outer_measure.trim = μ.to_outer_measure. This theorem shows that a restricted trimmed outer measure is a trimmed outer measure.