On Orthogonal Probability Measures1

Definitions. Let X be an arbitrary set, J^ a Borel-field of some subsets B of X, and 9ii(jQ the family of all probability measures defined on J¿, i.e. the totality of all countably additive, non-negative set functions m(B), BE/¿, for which m(X) = 1. Henceforth the word "measure" denotes an element of M(jQ) and the expression "set of measures" a subset of Vïl(jQ. Two measures m and m' have been called equivalent (notation: m~m', as for example in [l]), if m'(B) =0 for every set B in «£ for which m(B) = 0 and m(B) = 0 for every set B in «£ for which m'(B) — 0. A corresponding notion for sets of measures has been defined in [3] by Halmos and Savage in the following way: Two sets of measures M= \m\ and M'= {m'\ are called equivalent (notation: M~M'), it whenever m(B) =0 for all m in M, B in „£, then m'(B) =0 for all m' in M' and, conversely, whenever m'(B) =0 for all m' in M', B in /¿, then m(B)=0 for all m in M. Further, Halmos and Savage call a set of measures M={m] dominated (notation: M<p), if there exists a measure p such that m(B) =0 for every m in M, whenever p(B)=0, BEJÇj Having established this terminology, HalmosSavage prove tbe following lemma, which will be used later: