Does there exist the Lebesgue measure in the infinite-dimensional space?

Abstract

We study the sigma-finite measures in the space of vector-valued distributions on the manifold $X$ with Laplace transform $$\Psi(f)=\exp\{-\theta\int_X\ln||f(x)||dx\}, \theta>0.$$ We also consider the weak limit of Haar measures on the Cartan subgroup of the group $SL(n,{\Bbb R})$ when $n$ tends to infinity. The measure in the limit is called {\it infinite dimensional Lebesgue measure}. It is invariant under the linear action of some infinite-dimensional Abelian group which is an analog of Cartan subgroup. The measure also is closely related to the Poisson--Dirichlet measures well known in combinatorics and probability theory. The only known example of the analogous asymptotical behavior of the uniform measure on the homogeneous manifold is {\it classical Maxwell-Poincar\'e lemma} which asserts that the weak limit of uniform measures on the Euclidean sphere of appropriate radius as dimension tends to infinity is the standard infinite-dimensional Gaussian measure and white noise, but in our situation all the measures are no more finite but sigma-finite.