Fatou's Lemma

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. For a sequence $(a_{m})_{m\in\mathbb{N}}$ in $[0,\infty]$, a sequence in $[-\infty,\infty]$ by The Extended Real Line, Its Order, and [0,∞] (ii), $\liminf_{m}a_{m}$ is its limit inferior of Tail Extrema, Limit Inferior and Limit Superior of a Sequence in [−∞,∞] (ii), the supremum over $k\in\mathbb{N}$ of its tail infima $\inf_{m\ge k}a_{m}$ of Tail Extrema, Limit Inferior and Limit Superior of a Sequence in [−∞,∞] (i). Let $(f_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{M}^{+}$, and let $g:X\to[0,\infty]$ be a map with $g(x)=\liminf_{m}f_{m}(x)$ for every $x\in X$.

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

(ii) $\int_{X}g\,d\mu\le\liminf_{m}\int_{X}f_{m}\,d\mu$.