Groups of Transformations without Finite Invariant Measures Have Strong Generators of Size 2

Abstract

A size 2 generator of a measure space $(\mathbf{X}, \mathscr{F}, p)$ under a set of $\mathbf{S}$ of transformation of $X$ is a partition $\{A, A^c\}$ of $X$ such that $\mathscr{F}$ is the smallest $\sigma$-algebra containing $\{s^{-1}A: s\in S\}$ up to sets of $p$-measure zero. Let $S$ be a semigroup of invertible nonsingular measurable transformations on a separable measure space $(X, \mathscr{F}, p)$ with $p(X) = 1$. Suppose that $S$ does not preserve any finite invariant measure absolutely continuous with respect to $p$. Then $\mathscr{F}$ has a size 2 generator $\{A, A^c\}$ and the orbit of $A$ under $S$ is dense in $\mathscr{F}$.