On the cohomology of measurable sets

Abstract

If T is an ergodic automorphism of a Lebesgue probability space (X,A,m), the set of coboundries B = db =T(b)+b with symmetric difference + form a subgroup of the set of cocycles A. Using tools from descriptive set theory, Greg Hjorth showed in 1995 that the first cohomology group H=A/B is uncountable. This can surprise, given that in the case of a finite ergodic probability space, H has only 2 elements. Hjorth's proof used descriptive set theory in the complete metric space (A,d(a,b)=m(a+b)), leading to the statement that B is meager in A. We use a spectral genericity result of Barry Simon to establish the same. It leads to the statement noted first by Karl Petersen in 1973 that for a generic a in A, the induced system T_a is weakly mixing, which is slightly stronger than a result of Nate Friedman and Donald Ornstein about density of weakly mixing in the space of all induced systems T_a coming from an ergodic automorphism T.