For $f_n$ non-negative, if $f_n \to f$ in measure, then $\int f \leq \lim \inf_{n \to \infty} \int f_n$

Question

This question is answered in If $f_n \geq 0$ and $f_n \to f$ in measure, then $\int f \leq \liminf \int f_n$.

I still have one question, why do we need to obtain a subsubsequence (denoted as $f_{n_{k_j}}$ in the linked question)? Why is working with the subsequence $f_{n_k}$ not enough?

I know this would be better as a comment on the previous question, but it is old so I doubt I will get a response at all.

Answer 1

Convergence in measure does not imply almost sure convergence. Since Fatou's Lemma requires almost sure convergence it is necessary to use the existence of a subsequence which converges almost surely.