Dichotomy of ergodic measures on linear spaces (Q1922335)

The note generalizes the paper of \textit{Y. Okazaki} [Ann. Inst. H. Poincaré, Probab. Stat. 21, 393-400 (1985; Zbl 0582.60004)], which proved a dichotomy theorem for \(H\)-ergodic Hausdorff space equipped with a cylindrical \(\sigma\)-algebra \(C(E,x^*_i)\), on which a measure \(\mu\) is defined. In the present paper, the author does not assume that the linear space is equipped with a topological structure, and the measurable space structure need not agree with the linear operations. Let \((E,{\mathcal E})\) be a measure space such that \(E\) is a linear set and the \(\sigma\)-algebra \(\mathcal E\) is invariant under translations by elements \(x\in E\), and let \(\mu\) and \(\nu\) be two \(H\)-ergodic probability measures on \(\mathcal E\), where \(H\subset E\). Then \(\mu\) and \(\nu\) are either equivalent or mutually singular. This result has a generalization: \(\mu\) may be \(H_\mu\)-ergodic and \(\nu H_\nu\)-ergodic, where \(H_\mu\neq H_\nu\), but in this case \(H_\mu\) and \(H_\nu\) are linear subspaces of \(E\).