If $(X_n)_n$ is a sequence of nonnegative random variable, $mathcal{Q}$ is a sub $sigma$-algebra, then almost surley $$E(liminf_n X_n|mathcal{Q}) leq liminf_n E(X_n|mathcal{Q}).$$

Let’s say that $(X_n)_n$ converges in probability to $X,$ is this true that, almost surely, $$E(X|mathcal{Q}) leq liminf_nE (X_n|mathcal{Q})$$