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Lebesgue's Series Theorem

Statement

Given a sequence of integrable functions $(g_n)$ such that $\sum_{n=1}^\infty \int |g_n| < \infty$, then $\sum_{n=1}^\infty g_n$ converges a.e. to an integrable function and $\int \sum_{n=1}^\infty g_n = \sum_{n=1}^\infty \int g_n$.

Proof

Use MCT for series.

all these pages adapted with probably insufficient credit from my university's lecture notes