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

This article is cited in 16 scientific papers (total in 17 papers)

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
Received in February 2007

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
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\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}
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2007
\vol 259
\pages 248--272
\crossref{https://doi.org/10.1134/S0081543807040153}
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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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. V. V. Kozlov, “The generalized Vlasov kinetic equation”, Russian Math. Surveys, 63:4 (2008), 691–726  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    5. Vershik A.M., “Invariant measures for the continual Cartan subgroup”, J. Funct. Anal., 255:9 (2008), 2661–2682  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    8. A. M. Vershik, N. V. Smorodina, “Nonsingular transformations of the symmetric Lévy processes”, J. Math. Sci. (N. Y.), 199:2 (2014), 123–129  mathnet  crossref  mathscinet
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Kondratiev Yu., Lytvynov E., Vershik A., “Laplace Operators on the Cone of Radon Measures”, J. Funct. Anal., 269:9 (2015), 2947–2976  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  isi
    14. Kozlov V.V., “On the equations of the hydrodynamic type”, Pmm-J. Appl. Math. Mech., 80:3 (2016), 209–214  crossref  mathscinet  isi  scopus
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref
    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  mathnet  crossref  crossref  mathscinet  isi
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