Monotone Convergence Theorem

Let $(X,\mathcal{F},\mu)$ be a measure space and let $(f_m)_{m\in\mathbb{N}}$ be a sequence of measurable functions $f_m:X\to[0,\infty]$ such that $f_m(x)\le f_{m+1}(x)$ for every $x\in X$ and every $m$. Define $f:X\to[0,\infty]$ pointwise by $f(x)=\sup_m f_m(x)$, the least upper bound in $[0,\infty]$ (equal to $\infty$ when the values are unbounded). Then $f$ is measurable, and

the integrals being those of Lebesgue Integral of a Nonnegative Measurable Function; equivalently, the nondecreasing sequence of integrals converges to $\int_X f\,d\mu$ in $[0,\infty]$.