A Question about the Boundedness of the Conditional Expectation of a Random Variable 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

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.
