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Does there exist a Lebesgue measure in the infinite-dimensional space? (English. Russian original) Zbl 1165.28003
Proc. Steklov Inst. Math. 259, 248-272 (2007); translation from Tr. Mat. Inst. Steklova 259, 256-281 (2007).
The article is devoted to Lebesgue measure analogs in infinite-dimensional spaces. For this purpose suitable properties of characteristic functionals of measures are used. A measure space $$X$$ isomorphic with ($$[0,1], m$$) is considered, where $$[0,1]$$ is the real segment and $$m$$ is the Lebesgue measure on it. Weak convergence of a sequence of $$\sigma$$-finite Borel measures $$\mu _n$$ on the cone $$K$$ of all finite positive discrete measures on $$X$$ is used. By the definition this means that the limit $$\lim_n {\hat \mu }_n(f)$$ converges, where $${\hat \mu }_n(f)$$ is the Laplace transform of $$\mu _n$$ and $$f$$ is a positive function on $$X$$. The Laplace transform is defined as $$\int_K \exp \{ - \int_X f(x) d\xi (x) \} d\mu (\xi ) = {\hat \mu }(f)< \infty .$$ Then $$\sigma$$-finite, $$\sigma$$-additive and finite on compact sets measures are considered. The article contains also a review of many works in this area. The theorem of A. Weil, stating that an existence of a positive quasi-invariant Borel measure on a topological group relative to all (left) shifts $$L_h$$, $$h\in G$$, implies that $$G$$ is locally compact, is mentioned. This construction of measures is applied to infinite dimensional generalized Lie groups. In particular, $$\sigma$$-finite measures in the space of vector-valued distributions on $$X$$ with the characteristic functional $$\Psi (f) = \exp \{ - \theta \int_X \ln \| f(x) \| dx \},$$ $$\theta >0$$, are studied. The collection of such measures constitutes a one-parameter semigroup relative to $$\theta$$. In the case of scalar distributions and $$\theta =1$$, this measure was called in the paper as the infinite-dimensional Lebesgue measure. This measure is also closely related with the Poisson-Dirichlet measures well known in combinatorics and probability theory. Possibly the only known example of analogous asymptotic behaviour of the uniform measure on the homogeneous manifold is the classical Maxwell-Poincaré lemma, which states, that the weak limit of uniform measures on the Euclidean spheres of appropriate radii, as dimension tends to the infinity, is the standard infinite-dimensional Gaussian measure. Ideas for subsequent studies and applications of these activities, for example, on homogeneous spaces of Lie groups are discussed.

##### MSC:
 28A33 Spaces of measures, convergence of measures 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 46G12 Measures and integration on abstract linear spaces
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