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So, if $E[|X|] < \infty$, then $\int_{|X|>N+1}|X|d\mathcal{P} \to 0$ as $N \to \infty$.

This is very intuitively true... but just wanna confirm if there is actually a proper theorem that says this?

Thank you!

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Monotone convergence theorem does the job. Or we can directly use the definition of Lebesgue integral: for a fixed $\varepsilon$, there is $Y=\sum_{i=1}^Ma_i\chi_{E_i}$, where $E_i$ are measurable sets, $a_i$ real numbers and $M$ an integer, such that $\mathbb E|X-Y|\lt \varepsilon$. Then $$\int_{\{|X|\gt N\}}|X|\mathrm dP\leqslant \varepsilon+\int_{\{|X|\gt N\}}|Y|\mathrm dP\leqslant \varepsilon+\sum_{i=1}^M|a_i|\cdot\mathbb P(E_i\cap \{|X|\gt N\}).$$

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