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

Anatoly Vershik
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Abstract

We study the sigma-finite measures in the space of vector-valued distributions on the manifold $X$ with Laplace transform $$Ψ(f)=\exp\{-θ\int_X\ln||f(x)||dx\}, θ>0.$$ 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-Poincaré 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.

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Anatoly Vershik. 2008-02-02. Does there exist the Lebesgue measure in the infinite-dimensional space?. https://arxiv.org/abs/math-ph/0703033

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