The projection of a Borel set need not be a Borel set, and the projection of a Lebesgue measurable set need not be Lebesgue measurable. My question is, is the projection of a Jordan measurable set always Jordan measurable?
I expect the answer is no, but are there are any counterexamples to it?
From Wikipedia, a bounded set in $\mathbb{R}^n$ is Jordan measurable if and only if its boundary is Lebesgue null. In particular, it must be Lebesgue measurable.
Now take any bounded nonmeasurable subset $V$ of the real line. Then ${V\times\{0\}}$ is Jordan measurable since its closure has Lebesgue outer measure $0$. But its projection into the first coordinate is not.