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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?

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    Ok I see why this works. But what about sets other than the B(φ)? These are measurable sets in the algebra on X, but surely different types occur in the sigma algebra. Is it trivial to see that this works for all meas. sets?2012-12-24

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