Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 27H / a

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- Solution a

Solution

Fatou lemma states that for any sequence

(f_{n})

of nonnegative measurable functions on a measure space,

\int n \to \infty lim inf f_{n} d μ \leq n \to \infty lim inf \int f_{n} d μ .

For the proof, define

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

Then

(g_{n})

is a nonnegative increasing sequence and

g_{n} ↑ lim inf_{k \to \infty} f_{k} .

The monotone convergence theorem therefore gives

\int lim inf_{k} f_{k} d μ = lim_{n \to \infty} \int g_{n} d μ .

For each

k \geq n

, we have

g_{n} \leq f_{k}

, so monotonicity of the Lebesgue integral yields

\int g_{n} d μ \leq in f_{k \geq n} \int f_{k} d μ .

Taking the limit in

n

proves the asserted inequality and the proof of Fatou lemma.

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