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content/Definitions/Measure Theory/Fatou's Lemma.md

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+# Definition
+Have [[Measure|measure]] $\mu$ on $X$, and $f_{n} : X \to [0, \infty]$ [[Measurable|measurable]]. Then $\int \lim_{ n \to \infty } \inf f_{n} \, d\mu \le \lim_{ n \to \infty } \inf \int f_{n} \, d\mu$
+
+> [!info] What is $\lim\inf$?
+> Definition of [[Infimum|infimum]] (it is basically the opposite of a [[Supremum|supremum]]).
+> 
+> $\{ x_{n} \} \subset [0, \infty]$
+> $\lim_{ n \to \infty }\inf x_{n} = \sup_{m}\inf_{n \geq m} x_{n}$
+> 
+> $\inf_{n \geq m} = y_{m} \leq y_{m+1} \leq \dots$
+