Riemann integral and homoemorphism

Question

I am wondering what happens if I have the following setup:

I have a homeomorphism: $\phi$ from the unit sphere to the unit cube. I know that the characteristic function of the unit sphere is Riemann integrable. Can I conclude that the characteristic function of the unit cube (result of composition) is Riemann integrable?

Answer 1

Without using additional information (such as compactness), no. There is a homeomorphism between $(0,1)$ and $\mathbb R$, yet only one of their characteristic functions is Riemann integrable.