----- > [!proposition] Proposition. ([[Fatou's Lemma]]) > Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]] and $f_{1},f_{2},\dots$ is a [[sequence]] of nonnegative [[measurable function|measurable functions]] on $X$. Define a function $f:X \to [0, \infty]$ by $f(x):= \liminf_{k \to \infty}f_{k}(x)$. Then $f$ is a [[measurable function]], [[limit inferior and limit superior|and]] $\int f \, d\mu \leq \liminf_{k \to \infty} \int f_{k} \, d\mu. $ ^proposition > [!note] Note. > The [[monotone convergence theorem for nonnegative measurable functions|monotone convergence theorem]] is *not* required to prove Fatou's Lemma, but it provides a clean way to do so. ^note > [!proof]- Proof. ([[Fatou's Lemma]]) > First we show $f$ is a [[measurable function]]. It is enough to show $f^{-1}(B) \in \Sigma$ for [[generator criterion for function measurability|generators]] of the form $B=(t, \infty]$. > > $\{ x : \liminf_{k} f_{k}(x) > t\}= \bigcup_{N=1}^{\infty} \bigcap_{k \geq N}^{}\{ x: f_{k}(x) > t \}$ > > Suppose $x \in$ LHS, i.e., that $x$ is such that $\liminf_{k}f_{k}(x)>t$. [[sequence|Then]] since the sequence of [[infimum|infimums]] is [[monotonic map|monotonically increasing]], there exists $N \in \mathbb{N}$ such that for all $k \geq N$, $f_{k}(x)>t$. Hence $x \in$ RHS. For the reverse inclusion, suppose $x \in$ RHS: there exists some $N \in \mathbb{N}$ for which $f_{k}(x)>t$ for all $k \geq N$. Then certainly $x$ is on LHS. > > > The [[sequence]] $(\inf_{m \geq k} f_{m}(x))_{k \in \mathbb{N}}$ is [[monotonic map|monotonically increasing]] for each $x$, converging pointwise to $f(x)$; thus by the [[monotone convergence theorem for nonnegative measurable functions|monotone convergence theorem]] we have $\lim_{k \to \infty} \int \inf_{m \geq k} f_{m} \, d\mu = \int f \, d\mu ,$ > where the notation $\inf_{m \geq k}f_{m}$ denotes the function $(x \mapsto \inf_{m \geq k}f_{m}(x))$. Clearly then for a fixed $k$ one has $f_{m} \geq \inf_{m\geq k}f_{m}$ for any $m \geq k$, whence monotonicity of the [[integral]] gives $\int f_{m} \, d\mu \geq \int \inf_{m \geq k} f_{m} \, d\mu \text{ for all }m \geq k.$ > Applying $\inf_{m \geq k}$ to the LHS preserves this inequality; then $\lim_{k \to \infty} \left( \inf_{m \geq k} \int f_{m} \, d\mu \right) \geq \lim_{k \to \infty} \int \inf_{m \geq k} f_{m} \, d\mu ,$ > from which the result follows. > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```