Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 27H (Probability and Measure)

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.

On the probability space $([0,1],\mathcal{B},\lambda )$, letFor every $x\in [0,1]$, eventually $x\notin (0,1/n)$, so $f_n(x)\to 0$. Nevertheless,for every $n$. ConsequentlyThis gives strict inequality in Fatou lemma.

Yes. Almost-sure convergence givesalmost surely. Since the random variables are nonnegative, Fatou lemma applies and givesThe inequalities remain valid in the extended nonnegative reals, so no integrability assumption beyond nonnegativity is required.