I have a set $S$ that is a nonmeasurable subset of $X=\{0,1\}^{\mathbb{N}}$ (with respect to the normed product measure on $X$.
Now let $g:X\to[0,1]$ be defined by $g(x)=\underset{n\in\mathbb{N}}{\sum}\frac{x_{n}}{2^{n+1}}$.
Why is $g[S]$ not Lebesgue measurable?