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Let $f:[0,1]\rightarrow [0,1]$ be a function. Let $E\subseteq [0,1]$, such that $\mu(E)>0$. What conditions on $f$ would guarantee that $f(E)$ is also measurable, and $\mu(f(E))>0$?

I guess that $f$ should be measurable and continuous bijection, but this is not the correct condition.

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    Do you mean, what condition would ensure that *for all* measurable $E$ with positive measure, $f(E)$ is measurable with positive measure? – 2017-02-07
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    Correct. That is more precise. – 2017-02-07
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    It would suffice for $f$ to be bijective such that $f^{-1}$ is absolutely continuous, but this is not necessary. E.g., $(2x-1)^2$ is not bijective but satisfies the condition. – 2017-02-07

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