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Given a random variable $X$ on the space $\Omega$ endowed with sigma algebra $\mathcal{A}$. Let $\mathcal{F} \subset \mathcal{A}$ be a sub-sigma algebra.

How to prove that for the expectation value we have

$$E[E[X∣\mathcal{F}]]=E[X]$$?

user267839
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    By definition: $E[E[X|\mathcal{F}]] = \int_{\Omega}E[X|\mathcal{F}] ,d P = \int_{\Omega} X ,d P = E[X]$ as $\Omega \in \mathcal{F}$. – Sayantan Apr 19 '18 at 11:05
  • Well I can add that by defintion $E[X|\mathcal{F}]$ is the unique $\mathcal{F}$-measurable variable such that for any $A\in \mathcal{F}$ we have $E[1_A.E[X|\mathcal{F}]]=E[1_A.X]$ Taking $A=\Omega$ you get Sayantan self explanatory proof. Regards – TheBridge Apr 19 '18 at 14:04

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