OurBigBook.comSite Source code

Proof of Fatou lemma

Set

g_{n} = in f_{k \geq n} f_{k} .

Then

0 \leq g_{n} ↑ lim inf_{k} f_{k}

. The monotone convergence theorem and

g_{n} \leq f_{k}

for every

k \geq n

give

\int lim inf_{k} f_{k} d μ = lim_{n} \int g_{n} d μ \leq lim_{n} in f_{k \geq n} \int f_{k} d μ = lim inf_{k} \int f_{k} d μ .

 Ancestors (7)

 Incoming links (1)

Solution