/*
author: sailKite, elevict
source: https://discord.com/channels/686053708261228577/702656734631821413/1164269429543149598
*/
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> [!theorem] **Theorem 1.4.49** (Dominated convergence theorem).
> Let $(X,B,\mu)$ be a measure space, and let $f_1, f_2, \dots: X \rightarrow C$ be a sequence of measurable functions that converge pointwise $\mu$-almost everywhere to a measurable limit $f:X \rightarrow C$. Suppose that there is an unsigned absolutely integrable function $G:X \rightarrow [0,+\infty]$ such that $|f_n|$ are pointwise $\mu$-almost everywhere bounded by $G$ for each $n$. Then we have
> $$\lim_{n \rightarrow \infty}\int_{X}f_nd\mu = \int_{X}f~d\mu$$