Measure Theory/Convergence theorems

Theorem (monotone convergence theorem):

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

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

Theorem (Fatou's lemma):

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

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

Proof: Note that, upon defining

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

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

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

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

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

and if we take the lim inf,

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