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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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