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Dominated convergence theorem

If measurable functions converge almost everywhere and are dominated by an integrable function, then integrals and L1 norms converge.

Dominated convergence theorem

Dominated Convergence Theorem: Let $(X,\Sigma,\mu)$ be a measure space and let $(f_n)_{n\ge 1}$ be a sequence of measurable functions $f_n:X\to\mathbb{R}$ (or $\mathbb{C}$ ) such that $f_n\to f$ almost everywhere . Suppose there exists a nonnegative L1 function $g$ such that $|f_n|\le g$ almost everywhere for all $n$ . Then $f$ is Lebesgue integrable and

\lim_{n\to\infty}\int_X f_n\,d\mu \;=\; \int_X f\,d\mu, \qquad\text{and}\qquad \lim_{n\to\infty}\int_X |f_n-f|\,d\mu \;=\; 0.

In particular, $f_n\to f$ in $L^1(\mu)$ (see convergence in Lp with $p=1$ ).

Together with monotone convergence and Fatou's lemma , this theorem is a core tool for interchanging limits with the Lebesgue integral . It is especially useful when pointwise convergence is available but uniform bounds are only in the integrable sense.