Invariant measure for the continual Cartan subgroup

Abstract

We construct and study the one-parameter semigroup of $\sigma$-finite measures ${\cal L}^{\theta}$, $\theta>0$, on the space of Schwartz distributions that have an infinite-dimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the group $SL(n,R)$. The parameter $\theta$ is the degree of homogeneity with respect to homotheties of the space, we prove uniqueness theorem for measures with given degree of homogeneity, and call the measure with degree of homogeneity equal to one the infinite-dimensional Lebesgue measure $\cal L$. The structure of these measures is very closely related to the so-called Poisson--Dirichlet measures $PD(\theta)$, and to the well-known gamma process. The nontrivial properties of the Lebesgue measure are related to the superstructure of the measure PD(1), which is called the conic Poisson--Dirichlet measure -- $CPD$. This is the most interesting $\sigma$-finite measure on the set of positive convergent monotonic real series.