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Assume you are given a probability space $ ( \Omega, \mathcal{ F}, P ) $, a bounded random variable $ X $ on $ ( \Omega, \mathcal{ F}, P) $, and a sub-$\sigma$-algebra $ \mathcal{A} $ of $ \mathcal{F} $.

Is it true that the conditional expectation $ E[X | \mathcal{A}] $ of $ X $ given $ \mathcal{A} $ is again a bounded random variable?

Thanks a lot for your help! Regards, Si

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up vote 2 down vote accepted

Yes. The result follows from the fact that, if $X_1\le X_2$ a.s., then $E(X_1|\mathcal{A})\le E(X_2|\mathcal{A})$.

Let $-B\le X\le B$ for some constant $B$ and apply the above result.

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Hi Ashok! Thanks a lot! That's what I was looking for! –  Mad Si Nov 21 '11 at 10:53

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