# 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

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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.