Question about the norm on L^p space

Question

I am trying to see how $L^p(E)$, the space of measurable functions that are p-Lesbgue integrable, is a normed space with the norm $$\Vert f\Vert:=(\int_E |f(x)|^pdx)^{\frac{1}{p}}$$. I know how to prove tringle inequality and homogenity, however I am getting stuck with the first axiom "which I feel bad about me for that". For a normed space $X$, it satisfies that $\Vert x \Vert=0$ iff $x=0$ but it does not say anything about almost every where "a.e". For a Lesbgue integrable function $f \in L^p(E)$, it is know that $\Vert f \Vert=f$ would imply that $f=0$ a.e on $E$ i.e $f \neq 0$ on $A \subset E$ with measure zero. How does not this contradict the definition of a normed space? I would apprecite any explanation about that.

Answer 1

You're right, this would be a problem! For this reason, we don't define $L^p$ to actually be the space of $p$-integrable functions. Instead, the elements of $L^p$ are equivalence classes of $p$-integrable functions, where two functions are equivalent when they are equal almost everywhere. This is a quotient of the vector space of $p$-integrable functions by the subspace of functions which are $0$ a.e., so it naturally inherits a vector space structure, and the norm you described is well-defined on this quotient.