On the existence of a finite invariant measure
This paper is a contribution to the study of existence of invariant measures not necessarily absolutely continuous with respect to a given measure. Let T be a Borel automorphism of a standard Borel space (X,\({\mathcal B})\) with measure m. Let sA denote the union of the sets \(T^ iA\quad (i\in {\mathbb{Z}}).\) Call \(A\in {\mathcal B}\) compressible if A is the disjoint union of sets, C, D with \(sC=sD=A\) such that A and C can be split into disjoint sets \(A_ i\quad (i\in {\mathbb{Z}}),\) resp. \(C_ i\quad (i\in {\mathbb{Z}}),\) of the form \(T^{n_ i}C_ i=A_ i.\) The main result states that if X is incompressible, then there exists a stochastic kernel \(\{m(A,x):\;x\in X,\quad A\in {\mathcal B}\}\) such that \((i)\quad m(A,x)=m(TA,x)=m(A,T,x),\) \((ii)\quad m(A,x)\equiv 0\) iff \(A\in {\mathcal H},\) and (iii) if \(TE=E,\) then for \(x\in E,\) \(m(A\cap E,x)=m(A,x).\) The kernel can be so chosen that the measures \(m(\cdot,x)\) yield all the ergodic invariant probability measures. (\({\mathcal H}\) is the \(\sigma\)-ideal of all sets A such that the union of the sets \(T^ iA\quad (i\in {\mathbb{Z}})\) is compressible.)
scientific article; zbMATH DE number 4212372