# Range of function measurable?

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).

• 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.
– user59671
May 9, 2013 at 12:37
• The image of a Borel measurable function under a Borel-measurable may fail to be Borel, but I have no easy example. May 11, 2013 at 8:34

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.