Equivalent \(\sigma\)-finite invariant measures (Q852803)

The paper describes a ratio condition that is necessary and sufficient for a \(\sigma\)-finite measure space \((X, \beta, m)\), with a non-singular, bi-measurable, ergodic transformation \(T\) onto itself, to have a \(T\)-invariant \(\sigma\)-finite measure equivalent to \(m\). \(T\) is assumed to admit no wandering set, or, is conservative, i.e., there is no set \(A\) such that \(X = \bigcup_{n = - \infty}^ {n = \infty}T ^n A\), with \(\{T ^ n A \}\) pairwise disjoint. The condition is closely related to the one Dowker obtained (1951) using a ratio convergence result in Hopf's ergodic theorem. To elaborate on the modified ratio, two examples are constructed, building an ergodic, non-singular transformation acting on a Lebesgue space.