mathlib documentation

measure_theory.content

Contents #

In this file we work with contents. A content λ is a function from a certain class of subsets (such as the the compact subsets) to ℝ≥0 that is

We show that:

We define bundled contents as content. In this file we only work on contents on compact sets, and inner contents on open sets, and both contents and inner contents map into the extended nonnegative reals. However, in other applications other choices can be made, and it is not a priori clear what the best interface should be.

Main definitions #

For μ : content G, we define

We prove that, on a locally compact space, the measure μ.measure is regular.

References #

structure measure_theory.content (G : Type w) [topological_space G] :
Type w

A content is an additive function on compact sets taking values in ℝ≥0. It is a device from which one can define a measure.

@[instance]

Although the to_fun field of a content takes values in ℝ≥0, we register a coercion to functions taking values in ℝ≥0∞ as most constructions below rely on taking suprs and infs, which is more convenient in a complete lattice, and aim at constructing a measure.

Equations
theorem measure_theory.content.mono {G : Type w} [topological_space G] (μ : measure_theory.content G) (K₁ K₂ : topological_space.compacts G) (h : K₁.val K₂.val) :
μ K₁ μ K₂
theorem measure_theory.content.sup_disjoint {G : Type w} [topological_space G] (μ : measure_theory.content G) (K₁ K₂ : topological_space.compacts G) (h : disjoint K₁.val K₂.val) :
μ (K₁ K₂) = μ K₁ + μ K₂
theorem measure_theory.content.sup_le {G : Type w} [topological_space G] (μ : measure_theory.content G) (K₁ K₂ : topological_space.compacts G) :
μ (K₁ K₂) μ K₁ + μ K₂

Constructing the inner content of a content. From a content defined on the compact sets, we obtain a function defined on all open sets, by taking the supremum of the content of all compact subsets.

Equations
theorem measure_theory.content.inner_content_of_is_compact {G : Type w} [topological_space G] (μ : measure_theory.content G) {K : set G} (h1K : is_compact K) (h2K : is_open K) :
μ.inner_content K, h2K⟩ = μ K, h1K⟩
theorem measure_theory.content.inner_content_mono {G : Type w} [topological_space G] (μ : measure_theory.content G) ⦃U V : set G⦄ (hU : is_open U) (hV : is_open V) (h2 : U V) :
μ.inner_content U, hU⟩ μ.inner_content V, hV⟩

This is "unbundled", because that it required for the API of induced_outer_measure.

theorem measure_theory.content.inner_content_Sup_nat {G : Type w} [topological_space G] (μ : measure_theory.content G) [t2_space G] (U : topological_space.opens G) :
μ.inner_content (⨆ (i : ), U i) ∑' (i : ), μ.inner_content (U i)

The inner content of a supremum of opens is at most the sum of the individual inner contents.

theorem measure_theory.content.inner_content_Union_nat {G : Type w} [topological_space G] (μ : measure_theory.content G) [t2_space G] ⦃U : set G (hU : ∀ (i : ), is_open (U i)) :
μ.inner_content ⋃ (i : ), U i, _⟩ ∑' (i : ), μ.inner_content U i, _⟩

The inner content of a union of sets is at most the sum of the individual inner contents. This is the "unbundled" version of inner_content_Sup_nat. It required for the API of induced_outer_measure.

theorem measure_theory.content.inner_content_mono' {G : Type w} [topological_space G] (μ : measure_theory.content G) ⦃U V : set G⦄ (hU : is_open U) (hV : is_open V) (h2 : U V) :
μ.inner_content U, hU⟩ μ.inner_content V, hV⟩

Extending a content on compact sets to an outer measure on all sets.

Equations
theorem measure_theory.content.outer_measure_eq_infi {G : Type w} [topological_space G] (μ : measure_theory.content G) [t2_space G] (A : set G) :
(μ.outer_measure) A = ⨅ (U : set G) (hU : is_open U) (h : A U), μ.inner_content U, hU⟩
theorem measure_theory.content.is_mul_left_invariant_outer_measure {G : Type w} [topological_space G] (μ : measure_theory.content G) [t2_space G] [group G] [topological_group G] (h : ∀ (g : G) {K : topological_space.compacts G}, μ (topological_space.compacts.map (λ (b : G), g * b) _ K) = μ K) (g : G) (A : set G) :
(μ.outer_measure) ((λ (h : G), g * h) ⁻¹' A) = (μ.outer_measure) A
theorem measure_theory.content.is_add_left_invariant_outer_measure {G : Type w} [topological_space G] (μ : measure_theory.content G) [t2_space G] [add_group G] [topological_add_group G] (h : ∀ (g : G) {K : topological_space.compacts G}, μ (topological_space.compacts.map (λ (b : G), g + b) _ K) = μ K) (g : G) (A : set G) :
(μ.outer_measure) ((λ (h : G), g + h) ⁻¹' A) = (μ.outer_measure) A
theorem measure_theory.content.outer_measure_pos_of_is_mul_left_invariant {G : Type w} [topological_space G] (μ : measure_theory.content G) [t2_space G] [group G] [topological_group G] (h3 : ∀ (g : G) {K : topological_space.compacts G}, μ (topological_space.compacts.map (λ (b : G), g * b) _ K) = μ K) (K : topological_space.compacts G) (hK : 0 < μ K) {U : set G} (h1U : is_open U) (h2U : U.nonempty) :

For the outer measure coming from a content, all Borel sets are measurable.

The measure induced by the outer measure coming from a content, on the Borel sigma-algebra.

Equations
@[instance]

In a locally compact space, any measure constructed from a content is regular.