Hi everybody. Can anyone help me to clarify these things? The definition of(adsbygoogle = window.adsbygoogle || []).push({}); F-measurable function is as this:

f:Ω→ℝ defined on (Ω,F,P) probability space isF-measurable if f^{-1}(B)={ω∈Ω: f(ω)∈B} ∈Ffor all B∈B(ℝ)

whereB(ℝ) is Borel field over ℝ and B is any Borel subset of the Borel field.

My confusions are:

1-Is the function f:Ω→ℝ 1-to-1?

2-Is f^{-1}(B):B(ℝ)→Fmapping to mutually exclusive and collectively exhaustive subsets ofF?

Thank you for any contributions.

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# Are F-measurable functions 1-to1?

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