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