If $X \in L_{p}(\Omega,\mathcal{F},P)$ for some $p \in [1,+\infty]$ how can I show that for the conditional expectation $Y = \mathbb{E}[X|\mathscr{p}]$ we have $$Y \in L_{p}(\Omega,\mathcal{F},P)$$
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What does the $\mathscr{p}$ in $Y = \mathbb{E}[X|\mathscr{p}]$ refer to ? Is it a sub-$\sigma$-algebra of $\mathcal{F}$ ? – Stratos supports the strike Jan 31 '21 at 22:28
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yes $\mathcal{F}$ is sub algebra of $\mathscr{p}$ – Jan 31 '21 at 22:40
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To make the notations clearer, I'll assume that $Y$ is the conditional expectation of $X$ given a sub-$\sigma$-algebra $\mathcal{G}$. Therefore we have $Y := \mathbb{E}[X|\mathcal{G}]$.
Now, if $p \in [1,+\infty[$, then the map $x \mapsto |x|^p$ is convex, and we can apply Jensen's Inequality for conditional expectation, therefore :
$$|Y|^p \le \mathbb{E}\left[|X|^p\,|\mathcal{G}\right] \tag{1} $$
Taking expectations of both sides in (1), we get
$$ \mathbb{E}\left[|Y|^p\right] \le \mathbb{E}\left[\mathbb{E}\left[|X|^p\,|\mathcal{G}\right]\right] =\mathbb{E}\left[|X|^p\right] $$
Hence, if $X \in L_{p}(\Omega,\mathcal{F},\mathbb{P})$, then $\mathbb{E}[X|\mathcal{G}] \in L_{p}(\Omega,\mathcal{F},\mathbb{P})$ too.
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The composition of a convex function ($x \mapsto |x|$) with a non-decreasing convex function ($x \mapsto x^p$) is convex. See e.g. this question for a proof. You can also refer to this question for other proofs that $x \mapsto |x|^p$ is convex – Stratos supports the strike Feb 08 '21 at 10:53