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 Tr. Mat. Inst. Steklova, 2007, Volume 259, Pages 256–281 (Mi tm579)

Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space?

A. M. Vershik

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: We consider sigma-finite measures in the space of vector-valued distributions on a manifold $X$ with the characteristic functional $\Psi(f)=\exp\{-\theta\int_XłnłVert f(x)\rVert dx\}$, $\theta>0$. The collection of such measures constitutes a one-parameter semigroup relative to $\theta$. In the case of scalar distributions and $\theta=1$, this measure may be called the infinite-dimensional Lebesgue measure. We prove that the weak limit of the Haar measures on the Cartan subgroups of the groups $\operatorname{SL}(n,\mathbb R)$, when $n$ tends to infinity, is that infinite-dimensional Lebesgue measure. This measure is invariant under the linear action of some infinite-dimensional abelian group that can be viewed as an analog of an infinite-dimensional Cartan subgroup; this fact can serve as a justification of the name Lebesgue as a valid name for the measure in question. Application to the representation theory of current groups was one of the reasons to define this measure. The measure is also closely related to the Poisson–Dirichlet measures well known in combinatorics and probability theory. The only known example of analogous asymptotic behavior 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 radius, as dimension tends to infinity, is the standard infinite-dimensional Gaussian measure. Our situation is similar, but all the measures are no more finite but sigma-finite. The result raises an important question about the existence of other types of interesting asymptotic behavior of invariant measures on the homogeneous spaces of Lie groups.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2007, 259, 248–272

Bibliographic databases:

UDC: 514.8

Citation: A. M. Vershik, “Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space?”, Analysis and singularities. Part 2, Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday, Tr. Mat. Inst. Steklova, 259, Nauka, MAIK «Nauka/Inteperiodika», M., 2007, 256–281; Proc. Steklov Inst. Math., 259 (2007), 248–272

Citation in format AMSBIB
\Bibitem{Ver07} \by A.~M.~Vershik \paper Does There Exist a~Lebesgue Measure in the Infinite-Dimensional Space? \inbook Analysis and singularities. Part~2 \bookinfo Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday \serial Tr. Mat. Inst. Steklova \yr 2007 \vol 259 \pages 256--281 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm579} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2433687} \zmath{https://zbmath.org/?q=an:1165.28003} \elib{http://elibrary.ru/item.asp?id=9572738} \transl \jour Proc. Steklov Inst. Math. \yr 2007 \vol 259 \pages 248--272 \crossref{https://doi.org/10.1134/S0081543807040153} \elib{http://elibrary.ru/item.asp?id=13548153} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-38849169948} 

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This publication is cited in the following articles:
1. A. M. Vershik, M. I. Graev, “Integral Models of Representations of Current Groups”, Funct. Anal. Appl., 42:1 (2008), 19–27
2. A. M. Vershik, M. I. Graev, “Integral Models of Unitary Representations of Current Groups with Values in Semidirect Products”, Funct. Anal. Appl., 42:4 (2008), 279–289
3. V. V. Kozlov, “The generalized Vlasov kinetic equation”, Russian Math. Surveys, 63:4 (2008), 691–726
4. Vershik A.M., “The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension”, J. Fixed Point Theory Appl., 3:2 (2008), 317–329
5. Vershik A.M., “Invariant measures for the continual Cartan subgroup”, J. Funct. Anal., 255:9 (2008), 2661–2682
6. A. M. Vershik, M. I. Graev, “Integral models of representations of the current groups of simple Lie groups”, Russian Math. Surveys, 64:2 (2009), 205–271
7. A. M. Vershik, M. I. Graev, “Poisson model of the Fock space and representations of current groups”, St. Petersburg Math. J., 23:3 (2012), 459–510
8. A. M. Vershik, N. V. Smorodina, “Nonsingular transformations of the symmetric Lévy processes”, J. Math. Sci. (N. Y.), 199:2 (2014), 123–129
9. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179
10. A. M. Vershik, M. I. Graev, “Cohomology in Nonunitary Representations of Semisimple Lie Groups (the Group $U(2,2)$)”, Funct. Anal. Appl., 48:3 (2014), 155–165
11. M. Bożejko, E. W. Lytvynov, I. V. Rodionova, “An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions”, Russian Math. Surveys, 70:5 (2015), 857–899
12. Kondratiev Yu., Lytvynov E., Vershik A., “Laplace Operators on the Cone of Radon Measures”, J. Funct. Anal., 269:9 (2015), 2947–2976
13. Hagedorn D., Kondratiev Yu., Lytvynov E., Vershik A., “Laplace Operators in Gamma Analysis”, Stochastic and Infinite Dimensional Analysis, Trends in Mathematics, eds. Bernido C., CarpioBernido M., Grothaus M., Kuna T., Oliveira M., DaSilva J., Birkhauser Boston, 2016, 119–147
14. Kozlov V.V., “On the equations of the hydrodynamic type”, Pmm-J. Appl. Math. Mech., 80:3 (2016), 209–214
15. V. Zh. Sakbaev, “Averaging of random walks and shift-invariant measures on a Hilbert space”, Theoret. and Math. Phys., 191:3 (2017), 886–909
16. A. M. Vershik, M. I. Graev, “Nonunitary representations of the groups of $U(p,q)$-currents for $q\geq p>1$”, J. Math. Sci. (N. Y.), 232:2 (2018), 99–120
17. D. V. Zavadsky, V. Zh. Sakbaev, “Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure”, Proc. Steklov Inst. Math., 306 (2019), 102–119
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