Problem:

Let $\{{X}_{i}{\}}_{i=1}^{\mathrm{\infty}}$ be a sequence of random variables on a probability space $(\mathrm{\Omega},\mathcal{F},P)$ such that $\underset{i\to \mathrm{\infty}}{lim}{X}_{i}=X\text{a.e.}$ a.e. Show that if $\underset{i}{sup}\text{E}({X}_{i}^{2})<\mathrm{\infty}$, then $\text{E}({X}^{2})<\mathrm{\infty}$.

My Attempt:

I will try to explain as best I can. First, I have a version of Fatou's Lemma stating that if $\{{X}_{i}{\}}_{i=1}^{\mathrm{\infty}}$ is a sequence of non-negative random variables, then $\text{E}(\underset{i}{lim\u2006inf}{X}_{i})\le \underset{i}{lim\u2006inf}\text{E}({X}_{i})$.

If we let ${Y}_{i}={X}_{i}^{2}$ then I have a sequence of non-negative random variables to work with. One concern of mine is this: can I assume that $\underset{i\to \mathrm{\infty}}{lim}{X}_{i}^{2}={X}^{2}$? I feel like that's necessary for for what I've written below to work.

If I can make that assumption then we have

$\begin{array}{rl}\text{E}({X}^{2})& =\text{E}(\underset{i}{lim\u2006inf}{X}_{i}^{2})\mathbf{\text{(Is this justified?)}}\\ & \le \underset{i}{lim\u2006inf}\text{E}({X}_{i}^{2})\text{(application of Fatou's Lemma)}\\ & \le \underset{i}{sup}\text{E}({X}_{i}^{2})\text{(property of real numbers)}\\ & \mathrm{\infty}\text{(by assumption)}.\end{array}$

Let $\{{X}_{i}{\}}_{i=1}^{\mathrm{\infty}}$ be a sequence of random variables on a probability space $(\mathrm{\Omega},\mathcal{F},P)$ such that $\underset{i\to \mathrm{\infty}}{lim}{X}_{i}=X\text{a.e.}$ a.e. Show that if $\underset{i}{sup}\text{E}({X}_{i}^{2})<\mathrm{\infty}$, then $\text{E}({X}^{2})<\mathrm{\infty}$.

My Attempt:

I will try to explain as best I can. First, I have a version of Fatou's Lemma stating that if $\{{X}_{i}{\}}_{i=1}^{\mathrm{\infty}}$ is a sequence of non-negative random variables, then $\text{E}(\underset{i}{lim\u2006inf}{X}_{i})\le \underset{i}{lim\u2006inf}\text{E}({X}_{i})$.

If we let ${Y}_{i}={X}_{i}^{2}$ then I have a sequence of non-negative random variables to work with. One concern of mine is this: can I assume that $\underset{i\to \mathrm{\infty}}{lim}{X}_{i}^{2}={X}^{2}$? I feel like that's necessary for for what I've written below to work.

If I can make that assumption then we have

$\begin{array}{rl}\text{E}({X}^{2})& =\text{E}(\underset{i}{lim\u2006inf}{X}_{i}^{2})\mathbf{\text{(Is this justified?)}}\\ & \le \underset{i}{lim\u2006inf}\text{E}({X}_{i}^{2})\text{(application of Fatou's Lemma)}\\ & \le \underset{i}{sup}\text{E}({X}_{i}^{2})\text{(property of real numbers)}\\ & \mathrm{\infty}\text{(by assumption)}.\end{array}$