Consider a locally compact group $G$, considered as a measurable space with the completed Borelstructure wrt. the Haarmeasure. Consider a map $f:G \mapsto to G$, which is measurable and has an inverse, which is then also measurable. Is $f$ an homeomorphism?
What if $G$ is abelian? If not, what are necessary conditions on $G$, such that this is the case.
For $\mathbb{R}$, it seems to be true: see http://math.stanford.edu/~chris/additive.pdf.