Dominated convergence theorem

Let $(X,\mathcal{A},\mu )$ be a measure space, let $f_n:X\to \mathbb{R}$ be measurable with $f_n\to f$ pointwise almost everywhere, and suppose there is a single $g\in L^1(\mu )$ with $\mathrm{\mid }{f}_n\mathrm{\mid }\le g$ almost everywhere for every $n$. Then $f$ and every $f_n$ are integrable,

${\lim }_{n\to \mathrm{\infty }}{\int }_X\mathrm{\mid }{f}_n-f\mathrm{\mid }d\mu =0,\text{hence}{\lim }_{n\to \mathrm{\infty }}{\int }_X{f}_{n}d\mu ={\int }_{X}fd\mu \mathrm{.}$

The domination hypothesis cannot be dropped: $f_n=n{\mathbf{1}}_{(0,1\mathrm{/}n)}$ on $[0,1]$ converges pointwise to $0$ while $\int f_{n}d\lambda =1$, and the least function dominating all $f_n$ is $1\mathrm{/}x$, which is not integrable on $(0,1)$.

Not proved in this library. It is recorded with citations and used in no proof here.

What would prove it. Fatou's lemma (Fatou's lemma ‡) applied to the nonnegative sequences $g+f_n$ and $g-f_n$, whose liminfs are $g+f$ and $g-f$; the two resulting inequalities squeeze $\int f_n$ to $\int f$. The convergence in $L^1$ comes from the same argument applied to $2g-\mathrm{\mid }{f}_n-f\mathrm{\mid }\ge 0$. So the three convergence theorems form one block: the monotone convergence theorem is proved from the construction of the integral, Fatou from monotone convergence, and this from Fatou.

Which page it serves. It is the endpoint of the Riemann integral page and of the uniform convergence page. Uniform convergence on a bounded interval is what this library can offer for interchanging a limit and an integral, and it is a much heavier hypothesis than pointwise convergence with a dominating function.

The Riemann-level substitute that is in scope. Arzela's bounded convergence theorem, that a uniformly bounded sequence of Riemann integrable functions on $[a,b]$ converging pointwise to a Riemann integrable limit may be integrated term by term, is a theorem about the Riemann integral and is not deferred. It needs the limit function's Riemann integrability as a hypothesis, which is exactly the weakness the Lebesgue theory removes.