Fatou lemma

For nonnegative measurable functions,

\int lim inf_{n} f_{n} d μ \leq lim inf_{n} \int f_{n} d μ .

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Fatou lemma for series

For nonnegative numbers

a_{n, k}

, Fatou lemma applied to counting measure gives

\sum_{k} lim inf_{n \to \infty} a_{n, k} \leq lim inf_{n \to \infty} \sum_{k} a_{n, k} .

Proof of Fatou lemma

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 .

Strict inequality in Fatou lemma

On

([0, 1], λ)

, the functions

f_{n} = n 1_{(0, 1/ n)}

converge pointwise to zero but satisfy

\int f_{n} d λ = 1

. Hence the two sides of Fatou lemma can be zero and one.

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