 Solution

For nonnegative measurable functions $f_n$ on a measure space, the Fatou lemma statesIn particular $\mathbb{E}[\mathrm{liminf}{X}_n]\le \mathrm{liminf}\mathbb{E}{X}_n$ for nonnegative random variables; infinite values are allowed. Nonnegativity, or a common integrable lower bound after subtraction, is essential.