# countably generated sigma algebra

Prove that X is a Random Variable IFF sigma field generated by X is countably generated.

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This community does not respond very well to commands. Perhaps you should include the work you've done so far? Otherwise it just seems like you're expecting us to do the work for you – Bey Nov 17 '10 at 17:31
Perhaps you misconstrued the "prove" as a command. My rationale for the succinctness was a respect for what seems like a "serious" math community. Hence zero formalities to save everyones time. And no I am not looking for anyone to spoon feed me the answers, hints would be much appreciated.If anyone is offended, my apologies. If it helps, Please read the question as "Please Prove...." – itsow Nov 17 '10 at 18:00

## 2 Answers

"If and only if" doesn't make sense here, but you should be able to prove that for any map $X:\Omega\to {\mathbb R}$ the $\sigma$-field generated by $X$, that is, $X^{-1}({\cal B}(\mathbb R))$, is countably generated. Hint: The $\sigma$-field of Borel sets of $\mathbb R$ is countably generated.

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Thanks Byron. Much appreciated! – itsow Nov 17 '10 at 18:00

Every countably generated sigma algebra is generated by a random variable. For a countable class A_i take the sum of f(I_(A_i))/10^k where f(x) is 4 when x=0 and 5 when x=1. Then the sigma algebra generated is the same as that generated by the countable class.

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Welcome to MSE! It really helps readability to format using MathJax (see FAQ). Regards – Amzoti Aug 11 '13 at 20:11