Let $(\Omega, \mathcal{A},P)$ be a probability space and let $X,Y\colon \Omega \rightarrow \mathbb{R}$ be random variables. Furthermore, let $Y$ be $p$-integrable. Then why is the conditional expectation $\mathbb{E}[Y \vert X]$ again necessarily $p$-integrable?
Kind regards and thank you very much!
By conditional Jensen's inequality and from the fact that $x\mapsto |x|^p$ is convex for $p\ge 1$, we find that $$ \left|\Bbb E\left[Y\big|X\right]\right|^p\le\Bbb E\left[|Y|^p\big|X\right],\quad\text{a.s.} $$ It follows that $$ \Bbb E\left[\left|\Bbb E\left[Y\big|X\right]\right|^p\right]\le \Bbb E\left[\Bbb E\left[|Y|^p\big|X\right]\right]=\Bbb E\left[|Y|^p\right]<\infty. $$