Monotone Convergence Theorem

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space, let $\mathcal{M}^{+}=\mathcal{M}^{+}(X,\mathcal{F})$ be the set of measurable maps into $[0,\infty]$ of Nonnegative Measurable Maps and Their Lebesgue Integral (i), and for $g\in\mathcal{M}^{+}$ let $\int_{X}g\,d\mu$ be its integral. Let $(f_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{M}^{+}$ with $f_{m}(x)\le f_{m+1}(x)$ for every $x\in X$ and every $m\in\mathbb{N}$, and let $f:X\to[0,\infty]$ be a map with $f(x)=\sup_{m}f_{m}(x)$ for every $x\in X$, the supremum as in Order and Suprema in the Extended Real Line (iii).

(i) $f\in\mathcal{M}^{+}$.

(ii) $\int_{X}f\,d\mu=\sup_{m}\int_{X}f_{m}\,d\mu$.