Sigma algebra generated by random variable on a set with generators

Cross Validated Asked by Gabriel on July 28, 2020

I’m having trouble proving an intuitive result I found in these lecture notes I’m using for self-study (1.2.14 there).

Suppose $X$ is a $(mathbb{S}, mathcal{S})$-valued random variable (from $(Omega, mathcal{F})$), and furthermore $mathcal{S} = sigma(mathcal{A})$. If $mathcal{F}^X$ is the $sigma$-algebra generated by $X$ in $Omega$, we want to show that $mathcal{F}^X = sigma({X^{-1}(A) : A in mathcal{A}})$.

It’s easy to prove that $mathcal{F}^X supset sigma({X^{-1}(A) : A in mathcal{A}})$, by noticing that (i) $mathcal{F}^X$ is a $sigma$-algebra, and that (ii) it contains ${X^{-1}(A) : A in mathcal{A}}$. But I believe I’m missing the right proof strategy for the other direction. Just appealing to the definitions and the tools developed so far (e.g. the $pi-lambda$ theorem) didn’t take me very far.

I think I get the spirit of the claim. Basically, it says that if you have a set of generators $mathcal{A}$ of $mathcal{S}$, to obtain $mathcal{F}^X$ you can either take the inverse images of all sets generated by $mathcal{A}$, or you can take the inverse images of just the sets in $mathcal{A}$ and then use those to generate a $sigma$-algebra. So, the order of the operations of "taking inverse images" and "generating a $sigma$-algebra" doesn’t matter. Is this understanding correct?

Any hint on a direction that might work for the proof would be extremely appreciated!

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