How to prove: ${X_n}$ Uniform absolute continuous and $X_n$ Convergence in probability to $X$ equivalent to $X$ in $L_p$?

Question

Help me my friends! Our teacher left a remark on the notes without providing a proof, which I am very curious about. It says that

If $\{|X_n|^p,n\geq1\}$ is uniform absolute continuous, and $X_n\xrightarrow{P}X$, then $$X\in L_p$$

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The definition of Uniform Absolute Continuous is provided as

Uniform absolute continuous: For all $\varepsilon$, there exists $\eta_{\varepsilon}$, such that for any $A \in \mathcal{A}$, if $\mathbb{P}(A)<\eta_{\varepsilon}$, then $$ \sup\limits_{i\in I}\mathbb{E}\left(|X_i|\mathbf{1}_A\right)<\varepsilon. $$

I know that with uniformly bounded moments, we can obtain uniform integrability from uniform absolute continuity, and hence I can easily prove that the r.v sequence is Lp convergent. However in this case, I do not have uniformly bounded moments.

From Fatou-Lebesgue Lemma, I can show that $$ \mathbb{E}|X|^p=\mathbb{E}\left[\liminf_{k\to\infty}|X_{n_k}|^p\right]\leq\liminf_{k\to\infty}\mathbb{E}|X_n|^p\leq\sup_n\mathbb{E}|X_n|^p. $$

But sadly this is how far I can get. How am I gonna prove that $\mathbb{E}|X_n|^p$ are all finite? Any helpful comments would be greatly appreciated!

Answer 1

There are some issues if the r.v takes the value of infinity with a positive probability. So the proof shown above is actually a little bit erroneous. Sorry I've been busy on my school work recently and didn't come back. If anyone wants a rigorous proof of the question, please refer to the following website, where I have posted a proof [labeld as Remark 2 in the file] written by my teacher. enter link description here

Answer 2

By going to a subsequence we may suppose $X_n \to X$ a.s. By Fatou's Lemma $E|X|^{p}1_{|X|>M} \leq \lim \inf_n E|X_n|^{p}1_{|X|>M}$. Choose $M$ such that $P(|X|>M)<\eta_{\epsilon}$ and apply the hypothesis to see that $E|X|^{p}1_{|X|>M}\leq \epsilon$. Of course, $E|X|^{p}1_{|X|\leq M}\leq M^{p}$.