Let $Y$ be a random variable s.t. $E|Y|<\infty$. Let $E[Y|\mathcal{G}]$ and $Y$ have the same distribution. I need to prove that $E[Y|\mathcal{G}]=Y$ a.s. How does one use the fact that a random variable and its conditional expectation has the same distribution ?
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"How does one use the fact that a random variable and its conditional expectation has the same value ?" Hopefully your proof won't use this "fact" since this is not the hypothesis. – Did Apr 14 '15 at 07:14
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@Did: sorry, same distribution, not value. – Anonymous Apr 14 '15 at 10:06