Lévy's upward theorem and $\mathcal{L}^p$ convergence. Lévy's upward theorem:
Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} \mathcal{F}_n)$.
Define $X_n = E[Y | \mathcal{F}_n]$ and $Z = E[Y | \mathcal{F}_{\infty}]$.
Then $X_n \to Z$ a.s. (and in $\mathcal{L}^1$)
Now for my problem suppose that $p > 1$ and $Y \in \mathcal{L}^p(\Omega, \mathcal{F}, P)$.
 How to show that $X_n \to Z$ in $\mathcal{L}^p$ ?
I thought about applying the dominated convergence theorem but I could not find a right bound for $|X_n - Z|^p$.
Any ideas? Thanks.
 A: Hints:


*

*Use Jensen's inequality in order to prove that $$\mathbb{E}(|X_n|^p) \leq \mathbb{E}(|Y|^p).$$

*Deduce from $$\sup_{n \geq 1} \mathbb{E}(|X_n|^p)< \infty$$ and Doob's maximal inequality that $$\sup_{n \geq 1} |X_n| \in L^p.$$

*Apply the dominated convergence theorem.

A: We have to show that if $ X\in\mathbb L^p$ and $X_n:=\mathbb E[X\mid \mathcal F_n]$, then $\mathbb E|X_n-\mathbb E[X\mid \mathcal F_\infty|^p\to 0$.
We can use a truncation argument: for a fixed $R$ we have
$$\mathbb E|X_n-Y|^p\leqslant 2^{p-1}\mathbb E\left|\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_n]-\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_\infty]\right|^p+\\
+2^{p-1}\mathbb E\left|\mathbb E[X\chi\{|X|\gt R\}\mid\mathcal F_n]-\mathbb E[X\chi\{|X|\gt R\}\mid\mathcal F_\infty]\right|^p.$$
Since $|\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_n]-\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_\infty]|$ is bounded by $2R$, we get 
$$\mathbb E|X_n-Y|^p\leqslant 2^{p-1}(2R)^{p-1}\mathbb E\left|\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_n]-\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_\infty]\right|\\
+4^{p-1}\mathbb E|X\chi\{|X|\gt R\}|^p.$$
Using the $\mathbb L^1$ case with $Y:=X \chi\{|X|\leqslant R\}$, we obtain 
$$\limsup_{n\to \infty}\mathbb E|X_n-Y|^p\leqslant 4^{p-1}\mathbb E|X\chi\{|X|\gt R\}|^p.$$
Since this bound is true for any $R$, we obtain the wanted conclusion by monotone convergence.
