Prove that X is a Random Variable IFF sigma field generated by X is countably generated.
countably generated sigma algebra
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probability
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6Perhaps 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...." – 2010-11-17
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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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0Thanks Byron. Much appreciated! – 2010-11-17
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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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0Welcome to MSE! It really helps readability to format using MathJax (see FAQ). Regards – 2013-08-11