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

A. M. Vershik

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 2, Tome 259 (2007), pp. 256-281.

Résumé

We consider sigma-finite measures in the space of vector-valued distributions on a manifold $X$ with the characteristic functional $\Psi(f)=\exp\bigl\{-\theta\int_X\ln\lVert f(x)\rVert\,dx\bigr\}$, $\theta>0$. 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.

Bibliographie

[1] Poincaré H., Calcul des probabilités, Gautiher-Villars, Paris, 1912 | Zbl

[2] Borel E., Introduction géométrique à quelques théories physiques, Gauthier-Villars, Paris, 1914

[3] Borel E., “Sur les principes de la théorie cinétique des gaz”, Ann. sci. Ecole Norm. Super. Sér. 3., 23 (1906), 9–32 | MR | Zbl

[4] Mehler F. G., “Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnung”, J. reine und angew. Math., 66 (1866), 161–176 | Zbl

[5] Maxwell J. C., “On Boltzmann's theorem on the average distribution of energy in a system of material points”, Trans. Cambridge Philos. Soc., 12 (1878), 547–570

[6] Cartier P., 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 | MR

[7] Stroock D. W., Probability theory: An analytic view, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl

[8] Yor M., Some aspects of Brownian motion. Pt. II: Some recent martingale problems, Birkhäuser, Basel, 1997 | MR | Zbl

[9] Diaconis P., Freedman D., “A dozen de Finetti-style results in search of a theory”, Ann. Inst. H. Poincaré. Probab. et Statist., 23:S2 (1987), 397–423 | MR | Zbl

[10] Yor M., Pitman J., “The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator”, Ann. Probab., 25:2 (1997), 855–900 | DOI | MR | Zbl

[11] Vershik A. M., “Izmerimye realizatsii grupp avtomorfizmov i integralnye predstavleniya polozhitelnykh operatorov”, Sib. mat. zhurn., 28:1 (1987), 52–60 | MR | Zbl

[12] Vershik A. M., “Opisanie invariantnykh mer dlya deistviya nekotorykh beskonechnomernykh grupp”, DAN SSSR, 218:4 (1974), 749–752 | Zbl

[13] Vershik A. M., “Klassifikatsiya izmerimykh funktsii neskolkikh argumentov i invariantno raspredelennye sluchainye matritsy”, Funkts. analiz i ego pril., 36:2 (2002), 12–27 | MR | Zbl

[14] Vershik A. M., Gelfand I. M., Graev M. I., “Predstavleniya gruppy $\operatorname{SL}(2,\mathbf R)$, gde $\mathbf R$ – koltso funktsii”, UMN, 28:5 (1973), 83–128 | MR

[15] Vershik A. M., Gelfand I. M., Graev M. I., “Kommutativnaya model predstavleniya grupp tokov $\operatorname{SL}(2,\mathbf R)^X$, svyazannaya s unipotentnoi podgruppoi”, Funkts. analiz i ego pril., 17:2 (1983), 70–72 | MR | Zbl

[16] Gel'fand I. M., Graev M. I., Vershik A. M., “Models of representations of current groups”, Representations of Lie groups and Lie algebras, ed. A. A. Kirillov, Akad. Kiado, Budapest, 1985, 121–179 | MR

[17] Vershik A. M., Graev M. I., “Kommutativnaya model predstavleniya gruppy $O(n,1)^X$ i obobschennaya lebegova mera v prostranstve raspredelenii”, Funkts. analiz i ego pril., 39:2 (2005), 1–12 | MR | Zbl

[18] Graev M. I., Vershik A. M., “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 (2005), 499–529 | DOI | MR | Zbl

[19] Tsilevich N., Vershik A., Yor M., “An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process”, J. Funct. Anal., 185:1 (2001), 274–296 | DOI | MR | Zbl

[20] Kingman J. F. C., Poisson processes, Clarendon Press, Oxford, 1993 | MR | Zbl

[21] Vershik A. M., Shmidt A. A., “Simmetricheskie gruppy vysokoi stepeni”, DAN SSSR, 206:2 (1972), 269–272 | MR | Zbl

[22] Vershik A. M., Shmidt A. A., “Predelnye mery, voznikayuschie v asimptoticheskoi teorii simmetricheskikh grupp. I, II”, Teoriya veroyatn. i ee prim., 22:1 (1977), 72–88 ; 23:1 (1978), 42–54 | MR | Zbl | MR | Zbl

[23] Ignatov Ts., “Ob odnoi konstante, voznikayuschei v asimptoticheskoi teorii simmetricheskikh grupp, i o merakh Puassona–Dirikhle”, Teoriya veroyatn. i ee prim., 27:1 (1982), 129–140 | MR | Zbl

[24] Vershik A. M., “Asimptoticheskoe raspredelenie razlozheniya naturalnykh chisel na prostye deliteli”, DAN SSSR, 289:2 (1986), 269–272 | MR

[25] Arnold V. I., “Vershik work needs acknowledgement”, Not. Amer. Math. Soc., 45:5 (1998), 568

[26] Tsilevich N. V., “Statsionarnoe raspredelenie sluchainogo razbieniya polozhitelnykh tselykh chisel”, Teoriya veroyatn. i ee prim., 44:1 (1999), 55–73 | MR | Zbl

[27] Diaconis P., Mayer-Wolf E., Zeitouni O., Zerner M. P. W., “The Poisson–Dirichlet law is the unique invariant distribution for uniform split–merge transformations”, Ann. Probab., 32 (2004), 915–938 | DOI | MR | Zbl

[28] Billingsley P., “On the distribution of large prime divisors”, Period. Math. Hungar., 2 (1972), 283–289 | DOI | MR | Zbl

[29] Arratia R., Barbour A. D., Tavaré S., Logarithmic combinatorial structures: A probabilistic approach, EMS Monogr. Math., Eur. Math. Soc., Zürich, 2003 | MR | Zbl

[30] Tenenbaum G., Introduction to analytic and probabilistic number theory, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[31] Yor M., “Some remarkable properties of gamma processes”, Advances in mathematical finance, Birkhäuser, Boston, 2007, 37–47 | MR | Zbl

[32] Vershik A., Yor M., 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] Vershik A. M., Tsilevich N. V., “Fokovskie faktorizatsii i razlozheniya prostranstv $L^2$ nad obschimi protsessami Levi”, UMN, 58:3 (2003), 3–50 | MR | Zbl

[34] fon Neiman Dzh., “Approksimativnye svoistva matrits vysokogo konechnogo poryadka”, Izbr. tr. po funktsionalnomu analizu, T. 1, Nauka, M., 1987, 277–327

[35] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, M., 1963 | MR

[36] Glasner E., Tsirelson B., Weiss B., “The autormorphism group of the Gaussian measure cannot act pointwise”, Israel J. Math., 148 (2005), 305–329 | DOI | MR | Zbl

[37] Kerov S., Olshanski G., Vershik A., “Harmonic analysis on the infinite symmetric group”, Invent. math., 158:3 (2004), 551–642 | DOI | MR | Zbl

[38] Veil A., Integrirovanie v topologicheskikh gruppakh i ego primeneniya, Izd-vo inostr. lit., M., 1950

[39] Vershik A. M., Graev M. I., “Integralnye modeli predstavlenii grupp tokov”, Funkts. analiz i ego pril., 42:1 (2008), 22–32 | MR | Zbl