Let $f: R^n \rightarrow R^m$ be any function. Will the graph of f always have Lebesgue measure zero ?

1) I could prove that this is true if f is continuous.

2) I suspect it is true if f is measurable, but I'm not sure. (My idea was to use Fubini's theorem to integrate the indicator function of the graph, but I don't know if I'm using the theorem properly).

If 2) is incorrect, what would be a counterexample where the graph of f has positive measure ?

If 2) is correct, can we prove the existence of a non-measurable function whose graph has positive measure ?