 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[Yes.] Almost-sure convergence gives
$$
X=\liminf_{n\to\infty}X_n
$$
almost surely. Since the <random variable>[random variables] are nonnegative, <Fatou lemma> applies and gives
$$
\mathbb E X
\leq\liminf_{n\to\infty}\mathbb E X_n
\leq\sup_{n\in\mathbb N}\mathbb E X_n.
$$
The inequalities remain valid in the extended nonnegative reals, so no integrability assumption beyond nonnegativity is required.

 Back to article page