Suppose $E \subset \mathbb R^d$ has measure $0$ and $f: \mathbb R^d \longrightarrow \mathbb R$ is measurable. Does $f (E)$ necessarily have measure $0$?
I tried to find a counter-example though I failed.It is clear that countable subset will not work for otherwise the image of it is again countable in $\mathbb R$ and hence has measure $0$. So we need to find an uncountable set of measure $0$ which maps to some set in $\mathbb R$ of positive measure under some measurable function in order to find a counter-example.How to find it?
Please help me in finding this example.Then it will really help me.
Thank you in advance.