I've been reading on Dominated Convergence Theorem and its proof using Fatou-Lebesgue, but I can't seem to figure out how to do so with Egorov's theorem.
If $\nu$ is a finite Baire measure on a compact Hausdorff space $X$, I first let $U_n$ be a sequence of sets such that $U_{n+1} \subset U_n$ with an additional property that $\nu(\cap_n U_n)=0$ .
I'm first trying to show that if $f_n$ is a characteristic function of $U_n$, then for any $g\in L^1$, $\lim_n \int | gf_n| \,d\nu=0$ using monotone convergence.
Any hints would be appreciated. Thank you.