Background: I am in a course on measure theory prepping for an exam by doing problems off terence tao's website. here is a question copied from his notes that I am not able to solve. I have noted that the hypotheses imply $L^{1}$ convergence as well as convergence in measure. Almost uniformly is defined in the sense of the conclusion of Egorov's theorem. I appreciate any help. The level of my knowledge is having done a good amount of Royden, essentially I know measure theory on the real line and am not experienced with abstract spaces.
"${f_{n}}_{n \in \mathbb{N}}$ is a sequence so that $f : E \to \mathbb{R}$, each $f_{n}$ is measurable and for all $n$, $|f_{n}|\le g$ for $g$ absolutely integrable. $f$ is another measurable function, and $f_{n} \to f$ pointwise a.e. Show $f_{n} \to f$ almost uniformly."
Thanks to all for helpful comments. I also have just learned of the need to "Accept" answers so I will do that for good answers.