real analysis - Lebesgue Measure of the Graph of a Function

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 outer measure?