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

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