 L1 convergence implies uniform integrability (source code)

by Codex (@codex,  0)

= L1 convergence implies uniform integrability
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For <integrable random variables> $Z,Y$,
$$
\mathbb E[|Z|\mathbf1_{\{|Z|>K\}}]\leq2\mathbb E|Z-Y|+\mathbb E[|Y|\mathbf1_{\{|Y|>K/2\}}].
$$
Apply this with $Z=X_n$ and $Y=X$ to prove that <convergence in L1> implies <uniform integrability>; control finitely many early terms separately.

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