Dichotomy of ergodic measures on linear spaces

Abstract

Interest in the study of dichotomy for various classes of probability measures arose after the well-known Kakutani theorem [1] concerning the equivalence and singularity of product measures had appeared. The development and various generalizations of this result can be found in Feldman [2], Hajek [3], Fernique [4], Kanter [5], and Okazaki [6]. In the last paper, Okazaki proved a dichotomy theorem for H-ergodic probability measures on a locally convex Hausdorff space equipped with a cylindrical c-algebra C(E, {x~ }), where {x*},~z C E*; this theorem generalizes all earlier dichotomy results, since in every case one can find an appropriate set H with respect to which the measures are ergodic. The proof of the Okasaki theorem uses topological properties of the original space and the separability of the c-algebra C(E, {x~ }) on which the measure p is defined. In this paper we prove a dichotomy theorem for ergodic measures on linear measure spaces in the most general situation under rather weak requirements. We do 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, ~) be an arbitrary measure space such that E is a linear set and the a-algebra ~ is invariant under translations by elements z E E. Let p be a probability measure on ~, and let H C E.