This one comes from Folland, Real Analysis, Problem 33 in the section titled Modes of Convergence.
Suppose $f_n \geq 0$ and $f_n \rightarrow f$ in measure, then $\int f \leq \liminf \int f_n$.
So I notice a few things first off, that since $f_n \to f$ in measure, we can find a subsequence $f_{n_j}$ which converges pointwise almost everywhere (Theorem 2.30 in Folland), and for this subsequence we may say (by Fatou's lemma using $f_n \geq 0$) that $\int f \leq \liminf \int f_{n_j}$, but it's not necessarily true that $\liminf \int f_{n_j} \leq \liminf \int f_n$, or at least I don't see how to prove it (and in general this is not true for any sequence and subsequence, while the reverse inequality is, I think).
Any tips, hints, or solutions?
Best Answer
You can pass to a subsequence $f_{n_k}$ with $\int f_{n_k} \to \liminf \int f_n$ first.
This subsequence will also converge to $f$ in measure and ... then you already know what to do.