Understanding Uniform Convergence: The Role of N and A

[EV33]

Homework Statement

I would just like to be pointed in the right direction. I have this 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 given ε>0 and $\delta$>0, there is a set A$\subset$E with mA<$\delta$, and an N such that for all x$\notin$A and all n≥N,
lf_n(x)-f(x)l<ε

To me it appears to be concluding:
Given ε>0 and $\delta$>0, there is a set A$\subset$E with mA<$\delta$, and an N such that for all x$\notin$A and all n≥N,
<f_n> converges uniformly to a real-valued function f on E~A.



I know that this isn't case but I don't see why. So my question is what would the conclusion of this theorem need to say, in terms of ε and $\delta$, so that <f_n> converges uniformly to a real-valued function f on E~A?


Thank you for your time.

 

[hamsterman]

The difference between definitions of uniform and pointwise convergence is really small. What you have there is really uniform convergence. Although I don't see why you need delta and A..
Anyway, the difference is that in uniform convergence N does not depend on x. Swapping the conditions "there exists N" and "for all x" would produce the definition of pointwise convergence.