On Sullivan’s invariant measure problem

Abstract

Sullivan has posed an invariant measure problem for which a positive answer is very plausible. It also seems highly plausible that hyperfinite AW*-factors are injective. Surprisingly, it turns out that one of these problems must have a negative solution. Specifically we show that either the hyperfinite Takenouchi-Dyer factor is not injective, or no G G -invariant category measure exists for any free, ergodic action of a countable nonamenable group G G on a perfect Polish space.