Are all real-valued continous functions in $\mathbb{R}^2$ on a compact space integrable?

Question

If you look at all continous functions on a compact intervall in $\mathbb{R}$ this statement is true. They are all integrable (proof). Is this true in $\mathbb{R^2}$ and higher dimensional spaces?

I can't think of a function where this is false but I'm not quite sure. I tried to apply Fubini's theorem to break it down to but therefore the functions needs to be integrable in the first place.

Is there an simple argument I missed?

Answer 1

Continuous functions on a compact set are bounded, and compact sets have finite Lebesgue measure, thus all such functions are integrable.