Look at theorem 2.1, much more general than what you want, here:

http://www.math.msu.edu/~shapiro/pub...s/nonmsble.pdf

There they talk of [0,1) modulo 1 instead of the circle group, and in the proof where it

says "k+1 distinct elements of G" it must be "k+1 distinct elements of H"

Tonio