TITLE: Why do we need to define Lebesgue spaces using equivalence classes?
QUESTION [11 upvotes]: When we define an $L^{p}$ space for  $1\leq p \leq \infty$, we say elements of this space are equivalence classes of functions which are equal almost everywhere and $$ \int|f|^{p} dx < \infty $$
Why can we not say elements are functions which satisfy $ \int|f|^{p} < \infty $ ?
I understand that if $g=f$  a.e. then $ ||f||_{L^{p}} = ||g||_{L^{p}} $ is this the reason for it? 
EDIT :
The reason for asking is because I am studying an optimal control of PDEs course which says we need to be careful when considering the PDE :
$ -\Delta y = f $ on $ \Omega  $ 
$ y=0 $ on $ \partial \Omega  $
...since we need to define what it means for $ y=0 $ on $\partial\Omega$,  since  $\partial\Omega$ has zero measure.

REPLY [18 votes]: In order for the $L^p$-norm to truly be a norm, it needs to be true that $\| f \|_{L^p} = 0 \implies f = 0$. But if $f$ is a measurable function and $\int |f|^p \, dx = 0$, we can only conclude that $f$ is zero almost everywhere.