Vershik, A. M. Invariant measures for the continual Cartan subgroup. (English) Zbl 1187.37007 J. Funct. Anal. 255, No. 9, 2661-2682 (2008). The goal of the paper is to introduce a family of \(\sigma\)-finite measures in the space of Schwartz distributions on the interval (or on the manifold) that is invariant with respect to a continual group of multiplicators and has a finite degree of homogeneity with respect to homotheties. The measure with degree of homogeneity equal to one is called the infinite-dimensional Lebesgue measure.Therefore one parameter semigroup of \(\sigma\)-finite measures \({\mathcal L}^\theta\), \(\theta> 0\) on the space of Schwartz distributions which have an infinite-dimensional Abelian group of linear symmetries is defined. This group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the group \(\text{SL}(n,\mathbb{R})\).The uniqueness theorem which says that \({\mathcal L}^\theta\) is unique, ergodic, positive \(\sigma\)-finite measure on the cone of positive Schwartz distributions that is finite on compact subsets, satisfies certain invariance property, has a fixed degree of homogeneity is proved. Reviewer: Zagorka Lozanov-Crvenković (Novi Sad) Cited in 5 Documents MSC: 37A15 General groups of measure-preserving transformations and dynamical systems 37A05 Dynamical aspects of measure-preserving transformations 46T12 Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds Keywords:continual Cartan subgroup; infinite-dimensional Lebesgue measure; conic Poisson-Dirichlet measures PDFBibTeX XMLCite \textit{A. M. Vershik}, J. Funct. Anal. 255, No. 9, 2661--2682 (2008; Zbl 1187.37007) Full Text: DOI arXiv References: [1] N.I. 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