On local properties of k-parameter semiflows of nonsingular point transformations
Let \((\vartheta_ t:\) \(t\in (0,\infty)^ k)\) be a measurable semigroup of nonsingular transformations of a \(\sigma\)-finite measure space and \(\{T_ t\}\) the semigroup of contractions in \(L_ 1\) with \(<T_ tf,g>=<f,g\circ \vartheta_ t>\) for \(f\in L_ 1,\quad g\in L_{\infty}.\) It is shown that \(\{T_ t\}\) is strongly continuous. \(\{T_ t\}\) is strongly continuous on \([0,\infty)^ k\) with \(T_ 0=id\) iff \(\mu (E)>0\) implies \(\mu (\vartheta_ t^{-1}E)>0\) for some \(t>0.\)
ergodic properties of k-parameter measurable semiflows
measurable semigroup of nonsingular transformations
scientific article; zbMATH DE number 3987609