TITLE: Example of two functions that are equal almost everywhere?
QUESTION [6 upvotes]: We shall say that two functions $f$ and $g$ defined on a set $E$ are equal almost everywhere, and write $f(x)=g(x)$ a.e $x\in E$, if the set $\{x\in E: f(x)\neq g(x)\}$ has measure zero.

I just can't wrap my brain around the fact that such functions exist! Certainly, we can take the cantor set which has measure zero but how to pick $f$ and $g$? Is there an example that I am not aware of ?

REPLY [4 votes]: There are good answers already, but I think this needs a visual.