Suppose that $X \in \mathcal{X}$ and $X\sim P$, is integrable, i.e., $\int |X| \mathrm{d}P<\infty$. Let $(X_1,\ldots,X_n)$ be a vector of $n$ i.i.d. instances of $X$. Hence, $X^n\sim P^{\otimes n}$. Consider $\{\mathcal{S}_n\}_{n\in \mathbb{N}}$, $\mathcal{S}_n \subseteq \mathcal{X}^n$, be a sequence of sets, such that \begin{align} \lim \limits_{n \rightarrow \infty}\int_{\mathcal{S}_n} \mathrm{d}P^{\otimes n}=0. \end{align} Then, is the following argument true? \begin{align} \lim \limits_{n \rightarrow \infty} \int_{\mathcal{S}_n} \frac{1}{n}\sum \limits_{i=1}^n |X_i| \mathrm{d}P^{\otimes n}=0. \end{align}
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