## A pair of measure theory exercise of a “constructive” flavor [closed]

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

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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 Not a suitable question for a research forum – Fernando Muro Feb 16 2012 at 23:25