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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 ennreal (or nnreal) that is

We show that:

We don't explicitly define the type of contents. 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

References

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.inner_content_le {G : Type w} [topological_space G] {μ : topological_space.compacts Gennreal} (h : ∀ (K₁ K₂ : topological_space.compacts G), K₁.val K₂.valμ K₁ μ K₂) (U : topological_space.opens G) (K : topological_space.compacts G) :

theorem measure_theory.inner_content_of_is_compact {G : Type w} [topological_space G] {μ : topological_space.compacts Gennreal} (h : ∀ (K₁ K₂ : topological_space.compacts G), K₁.val K₂.valμ K₁ μ K₂) {K : set G} (h1K : is_compact K) (h2K : is_open K) :
measure_theory.inner_content μ K, h2K⟩ = μ K, h1K⟩

theorem measure_theory.inner_content_mono {G : Type w} [topological_space G] {μ : topological_space.compacts Gennreal} ⦃U V : set G⦄ (hU : is_open U) (hV : is_open V) :

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

theorem measure_theory.inner_content_Sup_nat {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} (h1 : μ = 0) (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) (U : topological_space.opens G) :
measure_theory.inner_content μ (⨆ (i : ), U i) ∑' (i : ), measure_theory.inner_content μ (U i)

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

theorem measure_theory.inner_content_Union_nat {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} (h1 : μ = 0) (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) ⦃U : set G (hU : ∀ (i : ), is_open (U i)) :
measure_theory.inner_content μ ⋃ (i : ), U i, _⟩ ∑' (i : ), measure_theory.inner_content μ 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.inner_content_pos {G : Type w} [topological_space G] [t2_space G] [group G] [topological_group G] {μ : topological_space.compacts Gennreal} (h1 : μ = 0) (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) (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 : topological_space.opens G) :

theorem measure_theory.inner_content_mono' {G : Type w} [topological_space G] {μ : topological_space.compacts Gennreal} ⦃U V : set G⦄ (hU : is_open U) (hV : is_open V) :

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

Equations
theorem measure_theory.outer_measure.of_content_le {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} {h1 : μ = 0} (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) (h : ∀ (K₁ K₂ : topological_space.compacts G), K₁.val K₂.valμ K₁ μ K₂) (U : topological_space.opens G) (K : topological_space.compacts G) :

theorem measure_theory.outer_measure.le_of_content_compacts {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} {h1 : μ = 0} (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) (K : topological_space.compacts G) :

theorem measure_theory.outer_measure.of_content_eq_infi {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} {h1 : μ = 0} (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) (A : set G) :
(measure_theory.outer_measure.of_content μ h1) A = ⨅ (U : set G) (hU : is_open U) (h : A U), measure_theory.inner_content μ U, hU⟩

theorem measure_theory.outer_measure.of_content_interior_compacts {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} {h1 : μ = 0} (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) (h3 : ∀ (K₁ K₂ : topological_space.compacts G), K₁.val K₂.valμ K₁ μ K₂) (K : topological_space.compacts G) :

theorem measure_theory.outer_measure.of_content_exists_open {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} {h1 : μ = 0} (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) {A : set G} (hA : (measure_theory.outer_measure.of_content μ h1) A < ) {ε : ℝ≥0} :

theorem measure_theory.outer_measure.of_content_preimage {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} {h1 : μ = 0} (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) (f : G ≃ₜ G) (h : ∀ ⦃K : topological_space.compacts G⦄, μ (topological_space.compacts.map f _ K) = μ K) (A : set G) :

theorem measure_theory.outer_measure.is_left_invariant_of_content {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} {h1 : μ = 0} (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) [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) :

theorem measure_theory.outer_measure.of_content_pos_of_is_open {G : Type w} [topological_space G] [t2_space G] {μ : topological_space.compacts Gennreal} {h1 : μ = 0} (h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ K₂) μ K₁ + μ K₂) [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} :