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Consider a function $f:\mathbb{R}\to\mathbb{R}$.

If $f$ is Borel measurable then is the range $f(\mathbb{R})$ Borel measurable (or maybe Lebesgue measurable).

Likewise if $f$ is Lebesgue measurable instead then is the range $f(\mathbb{R})$ Borel measurable (or maybe Lebesgue measurable).

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  • $\begingroup$ I'm not sure but I think for a Lebesgue measurable function the range is chosen to be just Borel measurable; so that we get a wider range of measurable functions. But for Borel measurable functions it seems to be more reasonable that both domain and range are Borel. $\endgroup$
    – user59671
    May 9, 2013 at 12:37
  • $\begingroup$ The image of a Borel measurable function under a Borel-measurable may fail to be Borel, but I have no easy example. $\endgroup$ May 11, 2013 at 8:34

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The Cantor-Lebesgue function carries a set of measure zero to a set of positive measure, and thus certain measurable sets to nonmeasurable sets. You can modify this example to find a measurable $f$ with nonmeasurable range.

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