Understanding Almost Everywhere Convergence I want to better understand the following statement:
Assume $supp(f)=E=supp(f_n)$ for all $n$ with $m(E)<\infty$, $E$ measurable.

A sequence of measurable functions $\{f_n\}\rightarrow f$ almost everywhere on $E$.



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*Does this mean:  


$$\lim_{n\rightarrow\infty}f_n(x)=f(x) \text{ for }x\in B\subset E \text{ with } m(B)>0 $$
What I am confused is how this compares with the poin-wise convergence.
I understand $f=g$ almost everywhere on $E$, but I don't understand what is the exact definition for "convergence".

Convergence almost everywhere is not a type of convergence.  We can say "$f$ converges point-wise to $g$ almost everywhere", or "$f$ converges uniformly to $g$ almost everywhere" etc.  The "almost everywhere" is saying that convergence happens on the whole domain except for a set of points with measure $0$.
To give you an example, let's say $f$ and $g$ are defined on the real line $\mathbb{R}$.  Now imagine a case where "$f$ converges point-wise to $g$ at every point except $x=5$".  Well this means $f$ converges point-wise to $g$ almost everywhere, because the only place it doesn't converge is the set $\{ 5 \}$, which has $0$ length
