In showing that defines a norm on , I'm not so sure about checking the axiom
A textbook mentions defining an equivalence relation on space but I don't really get what that's all about.
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.