Give an example of a measurable space (Ω,F) and a function X: Ω → R such that X^2 : ω → X2(ω) is F-B(R)-measurable whereas X is not?
Give an example of a measurable space
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real-analysis
1 Answers
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Hint: a characteristic function (i.e. the indicator function) of a set $A$ is measurable iff $A$ is measurable. Take a non-$F$-measurable set (so, $F$ should not be the whole $2^{\Omega}$) and modify (affinely) its characteristic function to take values $-1$ and $1$. Then a square is $1$, so it is measurable as a constant function, while the original function is not measurable.