How do you rigorously explain the fact that $u \in L^p$ can be non defined over sets of measure 0? In all the definitions of $L^p(\Omega)$ spaces I have been given these are defined to be the set of functions $f: \Omega \to  \mathbb{R}$ whose norm $||\cdot||_{L^p}$ is finite. We define is as the quotient with the equivalence relation $f=g$ if and only if $f=g$ almost everywhere.
Now, the books I am dealing with say that $f$ can be non-defined on sets of measure $0$. But the definition of  function explicitly says $f(x)$ is defined on every $x \in \Omega$. 
Do we really relax this last condition to be "defined almost everywhere" or the books I am dealing with want to express that we cannot make sure what the value at a especific set of measure $0$ is as it can always be redefined (but it is defined)? 

It doesn't matter - the two versions of the definition give isometrically isomorphic spaces. Allowing functions to be undefined on a set of measure zero can be convenient, for example allowing us to refer to $f(x)=|x|^{-1/2}$ as an element of $L^1([-1,1])$ without having to define $f(0)$. Or allowing us to define $f=\lim f_n$ when the limit only exists almost everywhere, etc.
It's so clear that it doesn't matter that people do use the second version, or write as though they were using it, without every worrying about giving a precise statement of the second version of the definition. If I wanted to state that definition precisely I'd probably start like so:
If $\mu$ is a measure on $X$ then an almost function on $X$ is a function $f:X\setminus E\to\Bbb C$ for some set $E$ with $\mu(E)=0$. 
Then if you feel like it you can give definitions of the sum and product of two almost functions, the integral of an almost function, what it means for two almost functions to be equal almost everywhere, etc, finally defining $L^p$ as a certain space of equivalence classes of almost functions.
