Fatou's Lemma

Statement

Suppose $\{f_{n}\}$ is a sequence of non-negative measurable functions, $f_{n}:X\to [0,+\infty ]$. Then:

$\int \liminf _{n\rightarrow +\infty }f_{n}\leq \liminf _{n\rightarrow +\infty }\int f_{n}$.

Proof

Define $g_{n}:=\inf _{k\geq n}f_{k}$ for all $n\in \mathbb {N}$.

By definition, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \liminf_{n\rightarrow +\infty} f_n= \lim_{n\rightarrow +\infty} (\inf_{k\geq n}f_k)=\lim_{n\rightarrow +\infty} g_n} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_n\leq g_{n+1}, \forall n \in \mathbb{N} } , so by Monotone Convergence Theorem,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow +\infty} \int g_n=\int \lim_{n\rightarrow +\infty} g_n = \int \liminf_{n\rightarrow +\infty} f_n} .

Furthermore, by definition we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_n\leq f_n, \forall n \in \mathbb{N}} , implying that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int g_n\leq \int f_n } .

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow +\infty} \int g_n } exists, taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \liminf_{n\rightarrow +\infty} } of both sides yields:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \liminf_{n\rightarrow +\infty} f_n=\lim_{n\rightarrow +\infty} \int g_n = \liminf_{n\rightarrow +\infty} \int g_n \leq \liminf_{n\rightarrow +\infty} \int f_n} .