I was studying the answers to the following question: About two functions whose Lebesgue integral on all sets of a $\sigma-$algebra are equal. Now I am wondering how to interpret the sets $\{f>g\}$, $\{f=g\}$,... and so on.
In the case where $f$ and $g$ are continuous (and thus Borel-measurable), the case is clear and we can write for example $\{f>g\}=\{x\in X\colon f(x)>g(x)\}$.
But what about the case where $f,g\in L^p(X)$ are measurable? I learned that elements of $L^p$-spaces are equivalence classes of measurable and integrable functions. So how can we define such a set, when the values of some function $f$ in the equivalence class $[f]$ are arbitrary in certain (or even every) points? If I choose representatives $f$ and $g$ and try to determine the set $\{f>g\}$, can't it become a completely different set, if I choose new representatives?
I always become completely confused every time when it comes to $L^p$-functions and pointwise arguments. Can you give me some simple or robust intuition?
The meaning of $[f]>[g]$ is that $f,g$ are almost everywhere real valued, and almost everywhere we have $f(x)>g(x)$. This doesn't depend on the choice of the representatives, because if you change the representatives you will change the functions only on a set of measure zero.