How Do Egorov's Theorem and Pointwise Convergence Differ in Measure Theory?

[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 itexts 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.

 

[fresh_42]

Yes, the point is only whether the choice of $N$ depends on location $x$ or not. I can't see a difference neither, so you possibly have swapped quantors, and the difference from 2 to 1 is indeed uniformity.