Suppose $f_n$ is a sequence of non-negative measurable functions that converges in measure to a measurable function $f$. I am trying to show that
$$\int f\leq\liminf_n\int f_n$$
Using Fatou's lemma we know that
$$\int\liminf_n f_n\leq\liminf_n\int f_n$$
so it would be enough to show that $$f\leq\liminf_n f_n$$ but, is this even true? am I going in the right direction?