# Constructing a measurable function and (non-)measurable sets, each with a curious property

In my study, I encountered the following interesting exercises of a decidedly "constructive" flavor:

(1) Construct a bounded measurable function $f: \mathbb{R} \rightarrow \mathbb{R}$ that is not Riemann integrable on any interval $(a,b) \subseteq \mathbb{R}.$

(2) Construct a non-measurable (resp., measurable) set in $\mathbb{R}^2$ which has a measurable (resp., non-measurable) projection onto any line.

I have limited experience with doing constructions, but I still find I have difficulties in seeing how to start work on/ flesh out constructions (sensitive to the particulars of the problem), and was wondering if anyone visiting has a more finely-tuned sense for how the arguments related to the constructions would proceed. Any constructive input (no pun intended), would be greatly appreciated.

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For (2): Does "measurable" refer to Borel measurable or Lebesgue measurable? –  Michael Greinecker Feb 15 '12 at 9:37
"measurable" is an umbrella for Lebesgue measurable; the book I am using is explicit when we want to assume f is Borel measurable. –  Vulcan Feb 15 '12 at 16:42

## 1 Answer

For (1) you can use the characteristic function of the rationals $\chi_{\mathbb Q}$

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