a.e.-defined integrable functions on $X$.

Question

I'm reading the book Folland-Real Analysis. On page 54 after the proposition 2.23 it is written:

With this in mind, we shall find it more convenient to redefine $L^{1}(\mu)$ to be the set of equivalence classes of a.e.-defined integrable functions on $X$, where $f$ and $g$ are considered equivalence iff $f=g$ a.e.

I did not understand what he means by:

" a.e.-defined integrable functions on $X$"

If anyone can help me, I'll be grateful.

Answer 1

Quoted from Foland:

So, when Foland says that a function is defined "only" a.e. then he means that from the point of view of integration it is enough to define a function almost everywhere; that is, on null sets the functions do not have to be defined.