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
- additive: If
K₂are disjoint sets in the domain of
λ(K₁ ∪ K₂) = λ(K₁) + λ(K₂);
- subadditive: If
K₂are in the domain of
λ(K₁ ∪ K₂) ≤ λ(K₁) + λ(K₂);
- monotone: If
K₁ ⊆ K₂are in the domain of
λ(K₁) ≤ λ(K₂).
We show that:
- Given a content
λon compact sets, we get a countably subadditive map that vanishes at
∅. In Halmos (1950) this is called the inner content
- Given an inner content, we define an outer measure.
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.
measure_theory.inner_content: define an inner content from an content
measure_theory.outer_measure.of_content: construct an outer measure from a content
- Paul Halmos (1950), Measure Theory, §53
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.
This is "unbundled", because that it required for the API of
The inner content of a supremum of opens is at most the sum of the individual inner contents.
The inner content of a union of sets is at most the sum of the individual inner contents.
This is the "unbundled" version of
It required for the API of
Extending a content on compact sets to an outer measure on all sets.