Let $(X,\mathfrak{A}, \mu)$ be a measure space and $(f_n)_{n \in \mathbb{N}}$ a sequence of functions such that $f_n: X \to ([0,\infty])$ is measurable for all $n \in \mathbb{N}$ and $f_1 \geq f_2 \geq \dots$.
I am asked to show that if $\displaystyle\lim_{n \to \infty}\int\limits_{X}f_n d\mu = 0$ then $f_n$ converges to the function zero almost everywhere.
I also wonder if the other direction holds.
I am having trouble because I can't apply any of the convergence theorems that we've learned.
How about Fatou:
$$\int_X \liminf f_n \le \liminf \int_x f_n = 0.$$
Now use that the $f_n$ are decreasing.
The other direction is not true. Take $f_n =\displaystyle \frac{1}{n^2}\chi_{[n,\infty]}$.