Exercise 40 - Show in Egoroff's theorem, the hypothesis "$\mu(X)<\infty$" can be replaced by "$|f_n|\le g$ for all $n$, where $g \in L^1(\mu)$."
Egoroff's Theorem - Suppose $\mu(X)<\infty$, and $f_1, f_2, ...$ and $f$ are measurable complex-valued functions on $X$ such that $f_n \rightarrow f$ a.e. Then for every $\epsilon > 0$ there exists $E\subset X$ such that $\mu(E)<\epsilon$ and $f_n \rightarrow f$ uniformly on $E^c$.
The proof of the original Egoroff's theorem uses continuity from above, which requires the finiteness assumption. I tried to use the dominating $g$ to get some finiteness condition.
When $\mu$ is $\sigma$-finite, I showed the integral of $g$ is concentrated on some set of finite measure. But this does not seem to help proving the statement since the complement of this set still have infinite measure (if $X$ has).
Can anyone give me some hint on the problem? Thank you