How Many Non-Measurable Sets Exist?

[fourier jr]

are there lots of examples of non-measurable sets? the one that seems to be in most textbooks involves a type of addition mod an irrational number with equivalence classes, etc etc, which in some books is done geometrically as rotations of a circle through an irrational angle. that example was by vitali, but there is another one by edward van vleck where he constructs two subsets of (0,1) in a way that either the measure of one is 1 & the other has measure 0, or they both have what he called "upper measure" =1, & therefore they're both non-measurable. how many other ones are out there that I don't know about?

 

[g_edgar]

My favorite is due to Bernstein.

You construct (by transfinite induction) a set B so that both B and its complement meet every uncountable closed set.