A homeomorphism can map a measurable set onto a non-measurable set

A homeomorphism $[0,1]\to[0,2]$ can map a Lebesgue measurable set onto a non-measurable set.Suggest a slogan

There exist a homeomorphism $\psi\colon[0,1]\to[0,2]$ and a Lebesgue measurable set $Z\subseteq[0,1]$ such that $\psi(Z)$ is not Lebesgue measurable.