Linear Transformation of Quasi-Invariant Measures

In harmonic analysis of a real separable Hilbert space //, we often wish to require a nice measure \i, whose measure theoretical structure is closely connected with the topological structure of H. In this direction, we have already known that an important measure is not a measure lying on H but rather a continuous cylindrical measure lying on a nuclear extension of H. Moreover it will be turned out that if ^ is also //-quasi-invariant, then the convergence of linear functional in \i is identical with the strong convergence in //, (see Theorem 2.1). Therefore //"-continuous (cylindrical) and //-quasi-invariant measures are regarded as nice measures and are worth special interest. From now on, realizing H as /, we shall consider these measures on R, Rg'c: /dR°°. RJ is the set of all x = (xl9..., xn,...)eR°° such that x,t = 0 except finite numbers of n. The general description for RJ-quasi-invariant measures was given by Skorohod. In [13] he characterized them in terms of a partial independence of sub-a-fields. But this result does not directly lead a classification of I-continuous and I-quasi-invariant measures. In above classification, we identify \JL and \i! if these measures are equivalent with each other. So it is desirable to have a concept which is invariant on the equivalence classes. One of these concepts is the set A^ of admissible linear operators on J, (see Definition 3.1). It seems to the author that A^ is a natural concept and plays an effective role in this problem. (It will be turned out in Theorem 3.2 that