PROBABILITY AND MATHEMATIC.AL STATISTICS. CYLINIPRICAL MEASURES ON TOPOLOGICAL GROUPbS

Abstract. The concept of cylindrical measures on locally compact Abelian groups is discussed. It is proved that if the convolution of two cylindrical measures,u and v on G extends to a Radon measure, then there exists an element a belonging to the Bohr compactification of G such that both p*S, and v * & have extensions to Radon measures. 1;. Cyldsisal measures on locally compact Ahban group. Let G be a locally compact Abelian (LCA) group and let r = G * denote its dual group equipped with the standard topology of the uniform convergence on compact subsets of G. We shall use the addive notation for the group operation in G and the multiplicative notation for T. fd will denote algebraically the same group r, but considered with the discrete topology. Bohr compactification bG of G is defined as the dual of T,. There exists a natural continuous injection G into bG, and in this sense G c bG. Because (bG) * = rd and G* = r, the sets of characters of bG and G coincide. Cylindrical subsets of G. For every n-tuple y,,..., y, a function = ( y..., y,J forms a continuous homomorphism of G into the n-dimensional torus T". ay will denote the smallest a-field of subsets of G such that y is measurable, i.e. By = (y-I (B): BE BOREL(Tn)] go = @O(G) stands for the union of all By. go is a field of sets but in general it is not a a-field. By an analogy with the notion of cylindrical subsets of locally convex linear spaces elements of B " will be called cylindrical subsets of G. Following this analogy, a non-negative normed finitely additive function p: go-+ 10, 1) such that, for every y, p restricted to By is countably additive, is called a cylindrical measure (c.m.) on G. Obviously, the restriction of a probability Radon measure on G to 93 ' is a c.m. If a c.m. has any extension to a Radon measure, then it is unique. A c.m. having an extension to a Radon measure will be called a Radon c.m. It should be noted than even if G = R cylindrical sets and c.rn.3 in the J. R o s i ~ i s k i and C. RylI-Nardzewski above sense differ from the objects with the same name studied in the measure theory on locally convex topological vector spaces. This will be easy to see by bijective characteristics of c.m.'s discussed further on (sa