zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] H. Poincaré, Calcul des probabilités (Gautiher-Villars, Paris, 1912).
[2] E. Borel, Introduction géométrique à quelques théories physiques (Gauthier-Villars, Paris, 1914).
[3] E. Borel, ”Sur les principes de la théorie cinétique des gaz,” Ann. Sci. Ec. Norm. Super., Sér. 3, 23, 9–32 (1906). · JFM 37.0944.01
[4] F. G. Mehler, ”Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnung,” J. Reine Angew. Math. 66, 161–176 (1866). · ERAM 066.1720cj
[5] J. C. Maxwell, ”On Boltzmann’s Theorem on the Average Distribution of Energy in a System of Material Points,” Trans. Cambridge Philos. Soc. 12, 547–570 (1878).
[6] P. Cartier, ”Le calcul des probabilités de Poincaré,” Preprint IHES/M/06/47 (IHES, Bures-sur-Yvette, 2006), http://www.ihes.fr/PREPRINTS/2006/M/M-06-47.pdf
[7] D. W. Stroock, Probability Theory: An Analytic View (Cambridge Univ. Press, Cambridge, 1993). · Zbl 0925.60004
[8] M. Yor, Some Aspects of Brownian Motion, Part II: Some Recent Martingale Problems (Birkhäuser, Basel, 1997). · Zbl 0880.60082
[9] P. Diaconis and D. Freedman, ”A Dozen de Finetti-Style Results in Search of a Theory,” Ann. Inst. H. Poincaré, Probab. Stat. 23(S2), 397–423 (1987). · Zbl 0619.60039
[10] M. Yor and J. Pitman, ”The Two-Parameter Poisson-Dirichlet Distribution Derived from a Stable Subordinator,” Ann. Probab. 25(2), 855–900 (1997). · Zbl 0880.60076
[11] A. M. Vershik, ”Measurable Realizations of Groups of Automorphisms, and Integral Representations of Positive Operators,” Sib. Mat. Zh. 28(1), 52–60 (1987) [Sib. Math. J. 28, 36-43 (1987)]. · Zbl 0627.47012
[12] A. M. Vershik, ”Description of Invariant Measures for the Actions of Some Infinite-Dimensional Groups,” Dokl. Akad. Nauk SSSR 218(4), 749–752 (1974) [Sov. Math., Dokl. 15, 1396–1400 (1974)].
[13] A. M. Vershik, ”Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices,” Funkts. Anal. Prilozh. 36(2), 12–27 (2002) [Funct. Anal. Appl. 36, 93–105 (2002)]. · Zbl 1025.28010
[14] A. M. Vershik, I. M. Gel’fand, and M. I. Graev, ”Representations of the Group SL(2, R), Where R Is a Ring of Functions,” Usp. Mat. Nauk 28(5), 83–128 (1973) [Russ. Math. Surv. 28 (5), 87–132 (1973)]. · Zbl 0288.22005
[15] A. M. Vershik, I. M. Gel’fand, and M. I. Graev, ”Commutative Model of Representation of the Group of flows SL(2, R)X That Is Connected with a Unipotent Subgroup,” Funkts. Anal. Prilozh. 17(2), 70–72 (1983) [Funct. Anal. Appl. 17, 137–139 (1983)]. · Zbl 0522.46017
[16] I. M. Gel’fand, M. I. Graev, and A. M. Vershik, ”Models of Representations of Current Groups,” in Representations of Lie Groups and Lie Algebras, Ed. by A. A. Kirillov (Akad. Kiado, Budapest, 1985), pp. 121–179.
[17] A. M. Vershik and M. I. Graev, ”A Commutative Model of a Representation of the Group O(n,1)X and a Generalized Lebesgue Measure in the Space of Distributions,” Funkts. Anal. Prilozh. 39(2), 1–12 (2005) [Funct. Anal. Appl. 39, 81–90 (2005)]. · Zbl 1135.22018
[18] M. I. Graev and A. M. Vershik, ”The Basic Representation of the Current Group O(n,1)X in the L 2 Space over the Generalized Lebesgue Measure,” Indag. Math. 16(3–4), 499–529 (2005). · Zbl 1147.22013
[19] N. Tsilevich, A. Vershik, and M. Yor, ”An Infinite-Dimensional Analogue of the Lebesgue Measure and Distinguished Properties of the Gamma Process,” J. Funct. Anal. 185(1), 274–296 (2001). · Zbl 0990.60053
[20] J. F. C. Kingman, Poisson Processes (Clarendon Press, Oxford, 1993).
[21] A. M. Vershik and A. A. Shmidt, ”Symmetric Groups of High Degree,” Dokl. Akad. Nauk SSSR 206(2), 269–272 (1972) [Sov. Math., Dokl. 13, 1190–1194 (1972)]. · Zbl 0285.60015
[22] A. M. Vershik and A. A. Shmidt, ”Limit Measures Arising in the Asymptotic Theory of Symmetric Groups. I, II,” Teor. Veroyatn. Primen. 22(1), 72–88 (1977) [Theory Probab. Appl. 22, 70–85 (1977)]; Teor. Veroyatn. Primen. 23 (1), 42–54 (1978) [Theory Probab. Appl. 23, 36–49 (1978)]. · Zbl 0375.60007
[23] Ts. Ignatov, ”On a Constant Arising in the Asymptotic Theory of Symmetric Groups, and on Poisson-Dirichlet Measures,” Teor. Veroyatn. Primen. 27(1), 129–140 (1982) [Theory Probab. Appl. 27, 136–147 (1982)]. · Zbl 0495.60055
[24] A. M. Vershik, ”The Asymptotic Distribution of Factorizations of Natural Numbers into Prime Divisors,” Dokl. Akad. Nauk SSSR 289(2), 269–272 (1986) [Sov. Math., Dokl. 34, 57–61 (1987)].
[25] V. I. Arnold, ”Vershik Work Needs Acknowledgement,” Notices Am. Math. Soc. 45(5), 568 (1998).
[26] N. V. Tsilevich, ”Stationary Random Partitions of a Natural Series,” Teor. Veroyatn. Primen. 44(1), 55–73 (1999) [Theory Probab. Appl. 44, 60–74 (2000)].
[27] P. Diaconis, E. Mayer-Wolf, O. Zeitouni, and M. P. W. Zerner, ”The Poisson-Dirichlet Law Is the Unique Invariant Distribution for Uniform Split-Merge Transformations,” Ann. Probab. 32, 915–938 (2004). · Zbl 1049.60088
[28] P. Billingsley, ”On the Distribution of Large Prime Divisors,’ Period Math. Hung. 2, 283–289 (1972). · Zbl 0242.10033
[29] R. Arratia, A. D. Barbour, and S. Tavaré, Logarithmic Combinatorial Structures: A Probabilistic Approach (Eur. Math. Soc., Zürich, 2003), EMS Monogr. Math.
[30] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory (Cambridge Univ. Press, Cambridge, 1995). · Zbl 0880.11001
[31] M. Yor, ”Some Remarkable Properties of Gamma Processes,” in Advances in Mathematical Finance (Birkhäuser, Boston, 2007), pp. 37–47. · Zbl 1156.60030
[32] A. Vershik and M. Yor, ”Multiplicativite du processus gamma etetude asymptotique des lois stables d’indice {\(\alpha\)}, lorsque {\(\alpha\)} tend vers 0,’ Prepubl. 289 (Lab. Probab., Univ. Paris VI, 1995).
[33] A. M. Vershik and N. V. Tsilevich, ”Fock Factorizations, and Decompositions of the L 2 Spaces over General Lévy Processes,” Usp. Mat. Nauk 58(3), 3–50 (2003) [Russ. Math. Surv. 58, 427–472 (2003)]. · Zbl 1060.46056
[34] J. von Neumann, ”Approximative Properties of Matrices of High Finite Order,” Port. Math. 3, 1–62 (1942). · JFM 68.0029.02
[35] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Fizmatgiz, Moscow, 1963; Academic, San Diego, CA, 2000). · Zbl 0918.65002
[36] E. Glasner, B. Tsirelson, and B. Weiss, ”The Autormorphism Group of the Gaussian Measure Cannot Act Pointwise,” Isr. J. Math. 148, 305–329 (2005). · Zbl 1105.37006
[37] S. Kerov, G. Olshanski, and A. Vershik, ”Harmonic Analysis on the Infinite Symmetric Group,” Invent. Math. 158(3), 551–642 (2004). · Zbl 1057.43005
[38] A. Weil, L’intégration dans les groupes topologiques et ses applications (Hermann, Paris, 1940), Actual. Sci. Ind. 869. · Zbl 0063.08195
[39] A. M. Vershik and M. I. Graev, ”Integral Models of the Representations of the Current Groups,” Funkts. Anal. Prilozh. 42 (2008) (in press). · Zbl 1162.22019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.