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Dominated Convergence Theorem (DCT)

Statement

Let $(f_n)$ be a sequence of measurable functions where:

• $(f_n(x))$ converges a.e. to a limit $f(x)$
• $\exists g$, an integrable function such that $\forall n: |f_n(x)| \le g(x)$ a.e.

Then $\int f = \lim_{n \to \infty} \int f_n$.

Corollaries

Bounded convergence theorem: if, rather than the function $g$, there exists a constant $c$ such that $\forall n: |f_n(x)| \le c$ a.e., then $\int f = \lim_{n \to \infty} \int f_n$.

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