# Measure Theory/Convergence theorems

Theorem (monotone convergence theorem):

Let ${\displaystyle (\Omega ,{\mathcal {F}},\mu )}$ be a measure space, and let ${\displaystyle f_{n}:\Omega \to [0,\infty ]}$ be an ascending (that is, ${\displaystyle f_{n+1}\geq f_{n}}$ pointwise) sequence of non-negative functions, that converges pointwise to a function ${\displaystyle f:\Omega \to [0,\infty ]}$. Then

${\displaystyle \lim _{n\to \infty }\int _{\Omega }f_{n}d\mu =\int _{\Omega }fd\mu }$.

Theorem (Fatou's lemma):

Let ${\displaystyle (\Omega ,{\mathcal {F}},\mu )}$ be a measure space, and let ${\displaystyle f_{n}:\Omega \to \mathbb {R} _{\geq 0}}$ be a sequence of non-negative functions. Then

${\displaystyle \int _{\Omega }\liminf _{n\to \infty }f_{n}d\mu \leq \liminf _{n\to \infty }\int _{\Omega }f_{n}d\mu }$.

Proof: Note that, upon defining

${\displaystyle g_{N}:=\inf _{k\geq N}f_{n}}$,

that the sequence of functions ${\displaystyle g_{N}}$ is strictly ascending and converges pointwise to ${\displaystyle f}$ as ${\displaystyle N\to \infty }$. Hence, the monotone convergence theorem is applicable and we obtain

${\displaystyle \lim _{N\to \infty }\int _{\Omega }g_{N}d\mu =\int _{\Omega }fd\mu }$.

Now for each ${\displaystyle N\in \mathbb {N} }$, we have

${\displaystyle \int _{\Omega }g_{N}d\mu \leq \int _{\Omega }f_{N}d\mu }$,

and if we take the lim inf,

${\displaystyle \lim _{N\to \infty }\int _{\Omega }g_{N}d\mu =\liminf _{N\to \infty }\int _{\Omega }g_{N}d\mu \leq \liminf _{N\to \infty }\int _{\Omega }f_{N}d\mu }$. ${\displaystyle \Box }$