If $E$ is Jordan measurable with measure zero, then $\int _E f=0$?

Question

Is it true that if $E$ is a Jordan measurable set of measure zero, then $\int _E f=0 $ ? note that I'm talking about Riemann integral here. I managed to prove it when $E$ is compact: in that case I can take a finite amount of rectangles $R_i$ such that $E \subset \bigcup R_i $ and $\sum Vol(R_i)< \epsilon$, for every $\epsilon >0$ and then use Riemann sums on a rectangle that contains $E$. In the general case I might only have an infinite amount of rectangles that contains $E$, and in this case I'm stuck. How can I prove it?

Answer 1

It is true.

And a set must be bounded to be Jordan measurable, so you don't need to worry about proving 'the general case'.