If $(X_n)_n$ is a sequence of nonnegative random variables and $\mathcal{Q}$ is a sub $\sigma$-algebra, then the conditional Fatou lemma holds almost surely, $$E[\liminf_n X_n\mid\mathcal{Q}] \leq \liminf_n E[X_n\mid\mathcal{Q}].$$
Let's say that $(X_n)_n$ converges in probability to $X$. Is it true that, almost surely, $$E[X\mid\mathcal{Q}] \leq \liminf_nE [X_n\mid\mathcal{Q}] \text{ ?}$$
Best Answer
This is false.
For an easy counterexample, let $X_n\geq 0$ be any sequence of random variables with $\liminf_n X_n = 0$ which converges to $X = 1$ in probability. Then let $\mathcal{Q}$ be large enough so that the $X_n$ are $\mathcal{Q}$-measurable. The inequality in question then becomes $X \leq \liminf_n X_n,$ which is false almost surely.
For instance, this is the case when $X_n$ is any sequence of independent random variables supported in $\{0,1\}$ such that $P(X_n = 0) \to 0$ but $\sum P(X_n = 0) = \infty$.