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.