Monotone convergence theorem (Beppo Levi)

Let $(X, \mathcal{A}, \mu)$ be a measure space and let $f_1 \le f_2 \le f_3 \le \cdots$ be measurable functions $X \to [0, +\infty]$ with pointwise limit $f = \lim_n f_n$. Then $f$ is measurable and

$\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu,$

both sides being allowed the value $+\infty$. Equivalently, for measurable $g_n : X \to [0, +\infty]$,

$\int_X \sum_{n=1}^{\infty} g_n \, d\mu = \sum_{n=1}^{\infty} \int_X g_n \, d\mu.$

No integrability hypothesis and no dominating function are needed: monotonicity and nonnegativity are the whole hypothesis.

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

What would prove it. The Lebesgue integral of a nonnegative measurable function as a supremum over simple minorants (Lebesgue measure and the Lebesgue integral ‡), plus continuity from below of the measure, that is $\mu(\bigcup_n E_n) = \lim_n \mu(E_n)$ for an increasing sequence of measurable sets. The standard argument fixes a simple $0 \le s \le f$ and $c \in (0,1)$, applies continuity from below to $E_n = \{f_n \ge c\,s\}$, and lets $c \to 1$. The theorem is then used immediately to prove that the integral is additive, so it is not a corollary of the basic theory but part of its foundation.

Which page it serves. The Riemann integral page, where the corresponding statement is false without extra hypotheses: an increasing sequence of Riemann integrable functions on $[0,1]$ with bounded integrals may converge pointwise to a bounded function that is not Riemann integrable. That failure is exactly the defect this theorem repairs, and it is the honest motivation for building a measure track at all.

A naming warning. "Monotone convergence theorem" names two unrelated results, and this library uses the phrase for both. The one stated here is Lebesgue's, sharpened by Beppo Levi in 1906, and it is about integrals of functions on a measure space. The other is the elementary theorem that a monotone sequence of reals converges if and only if it is bounded, proved in A monotone sequence converges if and only if it is bounded ↗. Nothing on this page bears on that sequence theorem, and neither result is a special case of the other. A reader who wants "bounded monotone sequences converge" is in the wrong item, and neither result is a special case of the other.

Interchange results that this library does prove. Uniform convergence permits interchange for the Riemann integral, and so does Arzela's bounded convergence theorem, which is a Riemann-level result and is in scope here even though its natural home is next to dominated convergence. The deferral is of the measure-theoretic statements, not of every interchange theorem.