# What's the Difference Between

1. Dec 1, 2007

### phreak

1) Let $$E$$ be a measurable set of finite measure, and $$\{ f_n \}$$ a sequence of measurable functions that converge to a real-valued function $$f$$ a.e. on $$E$$. Then, given $$\epsilon$$ and $$\delta$$, there is a set $$A\subset E$$ with $$mA < \delta$$, and an $$N$$ s.t. $$\forall x\notin A$$ and $$\forall n \ge N$$, $$|f_n(x) - f(x)| < \epsilon$$.

2) Egorov's Theorem: Let $$E$$ be a measurable set of finite measure, and $$\{ f_n \}$$ a sequence of measurable functions that converge to a real-valued function $$f$$ a.e. on $$E$$. Then there is a subset $$A\subset E$$ with $$mA < \delta$$ s.t. $$f_n$$ converges to $$f$$ uniformly on $$E\setminus A$$.

Most texts prove #2 from #1, and I'm confused as to what the difference is. I always thought the definition of uniform convergence was that if $$\epsilon > 0$$ is given, we can choose an $$N$$ such that $$\forall n \ge N$$, $$|f_n(x)-f(x)| < \epsilon$$.

Sorry if this is a stupid question, but I can't seem to wrap my brain around it. Thanks for the help.