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

Fatou lemma states that for any sequence $(f_n)$ of nonnegative measurable functions on a measure space,

For the proof, defineThen $(g_n)$ is a nonnegative increasing sequence andThe monotone convergence theorem therefore givesFor each $k\ge n$, we have $g_n\le f_k$, so monotonicity of the Lebesgue integral yieldsTaking the limit in $n$ proves the asserted inequality and the proof of Fatou lemma.