An Aspect of Quasi-Invariant Measures on $\mathbf R^∞$

Translationally Invariant Measure on the Infinite Dimensional Vector Space

“…§ 2o Gaussian Measure and Stationary Product Proof of Proposition 2.1. As proved in [3], the stationary product measure \ JL is R^-ergodic. Let I be the permutation group on the set of all natural numbers N = {1, 2,...}.…”

“…Z Q consists of such a permutation a E I that satisfies o(i) = i except finite numbers of i eN. As shown in [3], the measure \JL is 2J 0 -ergodic. Now, we shall derive a contradiction assuming that ^ has an equivalent R5>-invariant cr-finite measure v. Since u~v and /i is I^-invariant and I 0 -ergodic, if v is £ 0 -invariant, then we have f.i -cv for some constant c>0.…”

“…First we consider the product of Lebesgue measure and uniform probability measures on R"x[-1/2, 1/2]°°, and in the limit of n-»oo we obtain the requested measure /i on R°°. We can easily show that \JL is Mg 3 -invariant, and a detailed study shows that ju is (/^-invariant. We can modify \JL to get an (J 2 )-invariant measure, and other modifications can give invariant measures with respect to a larger subspace of R°°.…”

“…We have many other quasi-invariant measures ( [3]), but the known examples including the measure g above have no equivalent invariant measure. Historically the translational quasi-invariance of probability measures has been discussed in detail ( [4]), while the study of translational invariance has been neglected, perhaps because of difficulties of infinite measures which have less connection with the theory of probability.…”

Approximations and the Linearity of the Shepp Space

“…Such a space first appeared in Shepp [7] as 2 ( √ ρ) where ρ is a probability density, and had been studied by many authors concerned with the absolute continuity of the translation of infinite product measures [5,8,9]. Chatterji and Mandrekar [1] gave an example of 2 ( √ ρ) which is not a linear space.…”