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starbright49ly

starbright49ly

Answered question

2022-05-21

Need help:
Let ( Ω , F , P ) be a probability space. Suppose f : R R is some function such that f ( X ) is measurable for every real valued random variable X. I am curious if f ( X ) is necessarily σ ( X )-measurable. I tried to conclude with Doob-Dynkin lemma but to do we would need f to be B ( R ) measurable. Does someone has an idea or is this is false in general?

Answer & Explanation

Fahrleine9m

Fahrleine9m

Beginner2022-05-22Added 11 answers

It is possible that f ( X ) is measurable for every random variable X but f is not Borel. For example, if F = P ( Ω ), then f ( X ) is always measurable regardless of f and X. Obviously, we cannot infer that f is a Borel function.
Kash Brennan

Kash Brennan

Beginner2022-05-23Added 4 answers

To complement the previous answer. Let ( Ω , F , P ) = ( R , 2 R , δ 0 ) . Let f : R R be the indicator of some non-Borel set S and note that f ( X ) is measurable for any X (as mentioned in Danny's post). Let X be the identity from R to R , the latter endowed with B ( R ) . Then, X 1 ( f 1 ( { 1 } ) ) = S σ ( X ) = B ( R ).

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