I like this example, which is as natural as can be an example with sets that are not Lebesgue measurable. Start from the Cantor function $f:[0,1]\rightarrow \mathbb{R}$, and consider $h(x):= x+f(x)$, which is a homeomorphism $[0,1]\rightarrow[0,2]$. On each interval on the complement of the Cantor set $C$ this functions is a translation. Therefore $|h([0,1]\setminus C)|=|[0,1]\setminus C|=1$. Thus $|h(C)|=|[0,2]\setminus h([0,1]\setminus C)|=1$. So there exists a non measurable subset $V$ of $ h(C)$; let $W$ be $h^{-1}(V)\subset C$. Finally, the homeomorphism $h^{-1}$ maps this Lebesgue measurable set $W$ into the non-measurable set $V$.

I like this example, which is as natural as can be an example with sets that are not Lebesgue measurable. Start from the Cantor function $f:[0,1]\rightarrow \mathbb{R}$, and consider $h(x):= x+f(x)$, which is a homeomorphism $[0,1]\rightarrow[0,2]$. On each interval on the complement of the Cantor set $C$ this functions is a translation. Therefore $|h([0,1]\setminus C)|=|[0,1]\setminus C|=1$. Thus $|h(C)|=|[0,2]\setminus h([0,1]\setminus C)|=1$. So there exists a non measurable subset $V$ of $ h(C)$; let $W$ be $h^{-1}(V)\subset C$. Finally, the homeomorphism $h$ maps this Lebesgue measurable set $W$ into the non-measurable set $V$.

Also note that any Lebesgue, non Borel set in $h(C)$ is mapped by the homeomorphism $h^{-1}$ into a Lebesgue, non Borel subset of $C$.