Fatou's Lemma

Let $(X,\mathcal{F},\mu)$ be a measure space and let $(f_m)_{m\in\mathbb{N}}$ be a sequence of measurable functions $f_m:X\to[0,\infty]$. For $x\in X$ define

where the infimum and supremum are taken in $[0,\infty]$ with the conventions of Measure, Measure Space, and Probability Measure, and define $\liminf_m a_m$ for a sequence $(a_m)$ in $[0,\infty]$ in the same way. Then $\liminf_m f_m$ is measurable, and