DECOMPOSITION USING MEASURABLE FUNCTIONS

It has been known for some time (although no proof has been published) that, given two Lebesgue-measurable sets A and B in R(n) that are equ idecomposable under isometries g(1),...,g(k) belonging to an amenable group, then the characteristic functions chi(A) and chi(B) can be deco mposed as sums of nonnegative Lebesgue-measurable functions f(i) and f (i)og(i)(-1) (i = 1,...,k), respectively. We give a simple direct proo f using a linear operator mapping the space of bounded functions onto L(infinity).