Suppose $\mu$ is sigma-finite measure on a space $X$, and $ f_{n}$ converge to $f$ almost everywhere. Show that there exists measurable sets $E_{n}$ where $\mu ( \cap E_{n}^{c}) = 0$ and $f_{n}$ converge uniformly to $f$ on each $ E_{n}$.
It seems we want to use egorov for this, where you have your sets of finite measure are $X_{n}$. So, $\forall n$ and for each $ X_{n}$ there is a measurable set $A_{n}$ with $ \mu (X_{n} - A_{n}) < {1}/{2^{n}} $
This way our sets can have $\mu ( \cap A_{n}^{c}) $ as small as we want but not zero. How can we make them zero?
Thank you