This could totally be a stupid question but I'm unsure: is a measure (ie positive, countable additiv

Kolten Conrad 2022-07-12 Answered
This could totally be a stupid question but I'm unsure: is a measure (ie positive, countable additive on a σ algebra, 0 for the empty set) actually a measurable function (wrt to the Borel-sigma algebra on R )?
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Answers (2)

esperoanow
Answered 2022-07-13 Author has 11 answers
For a measure space ( X , F , μ ) you have that
μ 1 ( ( , c ] ) = { A F : μ ( A ) c }
so you will need a σ-algebra defined in F to define the measurability of μ. Well, you can define this σ-algebra using μ, this will give an induced σ-algebra in F , and we can note it by σ ( μ ).

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2d3vljtq
Answered 2022-07-14 Author has 5 answers
Given a function, a sigma algebra on the domain and a sigma algebra on the codomain, you can ask whether the function is measurable with respect to the given sigma algebras. You have only specified a sigma algebra on the codomain, so we can not answer your question.
What probably confused you is the fact that the domain of a measure is a sigma algebra on some set, so we need to consider a sigma algebra on another sigma algebra - the domain of the measure - to determine whether it is measurable.

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