In measure theory I encountered Egorov's theorem which states that if ( X , S...

In measure theory I encountered Egorov's theorem which states that if $(X,\mathcal{S},\mu )$ is a measure space such that $\mu (X)<\mathrm{\infty }$ i.e. $\mu$ is a finite measure.If $(f_n)$ be a sequence of measurable functions on $X$ converging pointwise to $f:X\to \mathbb{R}$,then $f_n$ is almost uniformly convergent to $f$.
Now in the book the definition of almost uniform convergence is the following:
$f_n\to f$ almost uniformly if for each $ϵ>0$,there exists $E_{ϵ}\subset X$ such that $\mu (E_{ϵ})<ϵ$ and $f_n\to f$ uniformly on $X-E_{ϵ}$.
Now in some books I have seen that almost everywhere means outside a measure 0 set.
So the definition of almost uniformly convergent should have been $f_n\to f$ almost uniformly if $\mathrm{\exists }E\subset X$ such that $\mu (E)=0$ and $f_n\to f$ uniformly on $X-E$.
But unfortunately the definition is not so.In fact the latter condition is stronger.I want to know why the former is taken as a definition and what the problem with the latter one is.I want to understand where I am making mistake in understanding the word "almost".