In showing that defines a norm on , I'm not so sure about checking the axiom
iff .
A textbook mentions defining an equivalence relation on space but I don't really get what that's all about.
Drexel is right except that is already defined as the equivalence class of functions under the relation f ~ g iff almost everywhere.
The main reason for this is that if you are doing measure theory (e.g. probability) you essentially don't care what happens on null sets. You will notice that people will abuse the notation a lot, by for example saying let and consider f(x), but there are ways around this.