×

Beyond Haar and Cameron-Martin: the Steinhaus support. (English) Zbl 1420.22004

In many purposes, one needs a reference measure. In Euclidean spaces, one has Lebesgue measure and in locally compact groups, one has Haar measure. Motivated by a Steinhaus-like interior point property involving the Cameron-Martin space of Gaussian measure theory, the authors study a group theoretic analogue, the Steinhaus triple \((H,G,\mu)\), and construct a Steinhaus support, a Cameron-Martin-like subset \(H(\mu)\) in any Polish group \(G\) corresponding to “sufficiently subcontinuous” measure \(\mu\), in particular for Solecki-type reference measure.

MSC:

22A10 Analysis on general topological groups
43A05 Measures on groups and semigroups, etc.
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Arhangel’skii, A.; Tkachenko, M., Topological Groups and Related Structures (2008), World Scientific · Zbl 1323.22001
[2] Arutyunyan, L. M.; Kosov, E. D., Spaces of quasi-invariance of product measures, Funct. Anal. Appl., 49, 142-144 (2015) · Zbl 1342.46041
[3] Badora, R., On the Hahn-Banach theorem for groups, Arch. Math. (Basel), 86, 517-528 (2006) · Zbl 1106.46008
[4] Banach, S., Théorie des opérations linéaires, Monografie Mat., vol. 1 (1932), (in: “Oeuvres”, vol. 2, PWN, 1979), translated as ‘Theory of linear operations’, North Holland, 1978 · JFM 58.0420.01
[5] Banach, S., Sur l’équation fonctionnelle \(f(x + y) = f(x) + f(y)\), Fundam. Math.. (Oeuvres, vol. I (1967), PWN: PWN Warszawa), 1, 47-48 (1920), (commentary by H. Fast, p. 314) · JFM 47.0235.02
[6] Banaszczyk, W., Additive Subgroups of Topological Vector Spaces, Lecture Notes in Mathematics, vol. 1466 (1991), Springer · Zbl 0743.46002
[7] Baker, R. L., “Lebesgue measure” on \(R^\infty \), Proc. Am. Math. Soc., 113, 4, 1023-1029 (1991) · Zbl 0741.28009
[8] Baker, R. L., “Lebesgue measure” on \(R^\infty \). II, Proc. Am. Math. Soc., 132, 2577-2591 (2004) · Zbl 1064.28015
[9] Bartoszewicz, A.; Filipczak, M.; Filipczak, T., On supports of probability Bernoulli-like measures, J. Math. Anal. Appl., 462, 26-35 (2018) · Zbl 1392.60005
[10] Becker, H.; Kechris, A. S., The Descriptive Set Theory of Polish Group Actions, London Math. Soc. Lecture Notes, vol. 232 (1996), Cambridge University Press · Zbl 0949.54052
[11] Berlinet, A.; Thomas-Agnan, C., Reproducing-Kernel Hilbert Spaces in Probability and Statistics (2004), Kluwer · Zbl 1145.62002
[12] Berz, E., Sublinear functions on \(R\), Aequ. Math., 12, 200-206 (1975) · Zbl 0308.39004
[13] Bingham, N. H., Variants on the law of the iterated logarithm, Bull. Lond. Math. Soc., 18, 433-467 (1986) · Zbl 0633.60042
[14] Bingham, N. H.; Fry, J. M., Regression: Linear Models in Statistics (2010), Springer · Zbl 1245.62085
[15] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1989), Cambridge University Press, (1st ed. 1987) · Zbl 0667.26003
[16] Bingham, N. H.; Kiesel, R., Risk-Neutral Valuation. Pricing and Hedging of Financial Derivatives (2004), Springer, (1st ed. 1998) · Zbl 1058.91029
[17] Bingham, N. H.; Ostaszewski, A. J., Kingman category and combinatorics, (Bingham, N. H.; Goldie, C. M., Probability and Mathematical GeneticsSir John Kingman Festschrift. Probability and Mathematical GeneticsSir John Kingman Festschrift, London Math. Soc. Lecture Notes in Mathematics, vol. 378 (2010), CUP), 135-168 · Zbl 1208.26006
[18] Bingham, N. H.; Ostaszewski, A. J., Normed groups: dichotomy and duality, Diss. Math., 472, 138p (2010) · Zbl 1231.22002
[19] Bingham, N. H.; Ostaszewski, A. J., Category-measure duality: convexity, midpoint convexity and Berz sublinearity, Aequ. Math., 91, 801-836 (2017) · Zbl 1434.26002
[20] Bingham, N. H.; Ostaszewski, A. J., Beyond Lebesgue and Baire IV: density topologies and a converse Steinhaus-Weil theorem, Topol. Appl., 239, 274-292 (2018) · Zbl 1454.26002
[21] Bingham, N. H.; Ostaszewski, A. J., Additivity, subadditivity and linearity: automatic continuity and quantifier weakening, Indag. Math. (N. S.), 29, 687-713 (2018) · Zbl 1384.39015
[22] Bingham, N. H.; Ostaszewski, A. J., Variants on the Berz sublinearity theorem, Aequ. Math., 93, 351-369 (2019) · Zbl 1411.26005
[23] Bingham, N. H.; Ostaszewski, A. J., The Steinhaus-Weil property: its converse, Solecki amenability and subcontinuity · Zbl 1463.22002
[24] Birkhoff, G., A note on topological groups, Compos. Math., 3, 427-430 (1936) · Zbl 0015.00702
[25] Bogachev, V. I., Gaussian Measures, Math. Surveys & Monographs, vol. 62 (1998), Amer Math Soc. · Zbl 0938.28010
[26] Bogachev, V. I., Measure Theory, vol. I, II (2007), Springer · Zbl 1120.28001
[27] Bogachev, V. I., Differentiable Measures and the Malliavin Calculus, Math. Surveys & Monographs, vol. 164 (2010), Amer Math Soc. · Zbl 1247.28001
[28] Cameron, R. H.; Martin, W. T., Transformations of Wiener integrals under translations, Ann. Math. (2), 45, 386-396 (1944) · Zbl 0063.00696
[29] Cameron, R. H.; Martin, W. T., Transformations of Wiener integrals under a general class of linear transformations, Trans. Am. Math. Soc., 58, 184-219 (1945) · Zbl 0060.29104
[30] Cameron, R. H.; Martin, W. T., The transformation of Wiener integrals by nonlinear transformations, Trans. Am. Math. Soc., 66, 253-283 (1949) · Zbl 0035.07302
[31] Christensen, J. P.R., On sets of Haar measure zero in abelian Polish groups, (Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces, Jerusalem, 1972. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces, Jerusalem, 1972, Israel J. Math., vol. 13 (1972)), 255-260, (1973)
[32] Christensen, J. P.R., Topology and Borel Structure. Descriptive Topology and Set Theory With Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, vol. 10 (1974) · Zbl 0273.28001
[33] Conway, J. B., A Course in Functional Analysis, Graduate Texts in Mathematics, vol. 96 (1990), Springer, (1st ed. 1985) · Zbl 0706.46003
[34] Dalecky, Yu. L.; Fomin, S. V., Measures and Differential Equations in Infinite-Dimensional Space (1991), Kluwer
[35] Diestel, J.; Spalsbury, A., The Joys of Haar Measure, Graduate Studies in Mathematics, vol. 150 (2014), Amer. Math. Soc. · Zbl 1297.28001
[36] Dodos, P., The Steinhaus property and Haar-null sets, Bull. Lond. Math. Soc., 41, 377-384 (2009) · Zbl 1171.28006
[37] Driver, B. K.; Gordina, M., Heat kernel analysis on infinite-dimensional Heisenberg groups, J. Funct. Anal., 255, 2395-2461 (2008) · Zbl 1163.43005
[38] Engelking, R., General Topology (1989), Heldermann, (1st ed. 1977) · Zbl 0684.54001
[39] Fell, J. M.G.; Doran, R. S., Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles: Vol. 1 Basic Representation Theory of Groups and Algebras (1988), Academic Press · Zbl 0652.46050
[40] Fort, M. K., A unified theory of semi-continuity, Duke Math. J., 16, 237-246 (1949) · Zbl 0034.32601
[41] Fremlin, D. H., Measure Theory: Broad Foundations, vol. 2 (2001), Torres Fremlin · Zbl 1165.28001
[42] Fremlin, D. H., Measure Theory, vol. 4Topological Measure Spaces, Part One (2003), Torres Fremlin
[43] Fuchs, L., Infinite Abelian Groups, vol. 1 (1970), Academic Press, vol. 2 1973
[44] Gardner, R. J.; Pfeffer, W. F., Borel measures, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), North-Holland), 961-1043 · Zbl 0593.28016
[45] Gel’fand, I. M.; Vilenkin, N. Ya., Generalized Functions, vol. 4Applications of Harmonic Analysis (1964), Academic Press · Zbl 0136.11201
[46] Gikhman, I. I.; Skorokhod, A. V., Theory of Random Processes I, Grundlehren Math. Wiss., vol. 210 (1971), Springer: Izdat. Nauka: Springer: Izdat. Nauka Moscow, reprinted 2004:
[47] Gill, T.; Zachary, W., Functional Analysis and the Feynman Operator Calculus (2016), Springer · Zbl 1345.81001
[48] Girsanov, I. V., On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory Probab. Appl., 5, 285-301 (1960) · Zbl 0100.34004
[49] Gordina, M., Heat kernel analysis and Cameron-Martin subgroup for infinite dimensional groups, J. Funct. Anal., 171, 192-232 (2000) · Zbl 1041.46027
[50] Gordina, M.; Laetsch, T., A convergence to Brownian motion on sub-Riemannian manifolds, Trans. Am. Math. Soc., 369, 6263-6278 (2017) · Zbl 1372.60119
[51] Gowrisankaran, C., Radon measures on groups, Proc. Am. Math. Soc., 25, 381-384 (1970) · Zbl 0202.40601
[52] Gowrisankaran, C., Quasi-invariant Radon measures on groups, Proc. Am. Math. Soc., 35, 503-506 (1972) · Zbl 0261.28016
[53] Gowrisankaran, C., Semigroups with invariant Radon measures, Proc. Am. Math. Soc., 38, 400-404 (1973) · Zbl 0258.22003
[54] Gross, L., Abstract Wiener spaces, (Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, vol. II, Part 1 (1967), Univ. California Press), 31-42 · Zbl 0187.40903
[55] Gross, L., Abstract Wiener measure and infinite-dimensional potential theory, (Lectures in Modern Analysis and Applications, II. Lectures in Modern Analysis and Applications, II, Lecture Notes in Mathematics, vol. 140 (1970), Springer), 84-116 · Zbl 0203.13002
[56] Haar, A., Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Math. Ann., 34, 147-169 (1933) · Zbl 0006.10103
[57] Halmos, P. R., Measure Theory, Grad. Texts in Math., vol. 18 (1950), Springer: Van Nostrand · Zbl 0040.16802
[58] Heyer, H., Probability Measures on Locally Compact Groups, Ergebnisse Math., vol. 94 (1977), Springer · Zbl 0373.60011
[59] Heyer, H., Recent contributions to the embedding problem for probability measures on a locally compact group, J. Multivar. Anal., 19, 119-131 (1986) · Zbl 0614.60007
[60] Heyer, H.; Pap, G., On infinite divisibility and embedding of probability measures on a locally compact abelian group, (Infinite-Dimensional Harmonic Analysis III (2005), World Sci. Publ.), 99-118 · Zbl 1109.60008
[61] Hewitt, E.; Ross, K. A., Abstract Harmonic Analysis, vol. I, Grundlehren Math. Wiss., vol. 115 (1979), Springer, (1st ed. 1963)
[62] Ibragimov, I. A.; Rozanov, Y. A., Gaussian Random Processes, Applications of Mathematics, vol. 9 (1978), Springer, Translated from the Russian by A.B. Aries · Zbl 0392.60037
[63] Jacod, J.; Shiryaev, A. N., Limit Theorems for Stochastic Processes, Grund. Math. Wiss., vol. 288 (2003), (1st ed. 1987) · Zbl 1018.60002
[64] Janson, S., Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics, vol. 129 (1997), Cambridge University Press · Zbl 0887.60009
[65] Kakutani, S., Über die Metrisation der topologischen Gruppen, Proc. Imp. Acad. (Tokyo), 12, 82-84 (1936), (reprinted in [67], Vol. 1, 60-62) · JFM 62.1230.03
[66] Kakutani, S., On equivalence of infinite product measures, Ann. Math. (2), 49, 214-224 (1948), (reprinted in [67], Vol. 2, 19-29) · Zbl 0030.02303
[67] Kakutani, S., (Kallman, R. R., Selected Papers, vol. 1, 2 (1986), Birkhäuser)
[68] Kallianpur, G., Zero-one laws for Gaussian processes, Trans. Am. Math. Soc., 149, 199-211 (1970) · Zbl 0234.60032
[69] Kaplansky, I., Infinite Abelian Groups (1969), U. Michigan Press, 1954 · Zbl 0194.04402
[70] Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156 (1995), Springer · Zbl 0819.04002
[71] Kemperman, J. H.B., A general functional equation, Trans. Am. Math. Soc., 86, 28-56 (1957) · Zbl 0079.33402
[72] Klee, V., Invariant extensions of linear functionals, Pac. J. Math., 4, 37-46 (1954) · Zbl 0055.10001
[73] Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality (1985), Birkhäuser: PWN: Birkhäuser: PWN Warszawa · Zbl 0555.39004
[74] Kuo, H. H., Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, vol. 463 (1975), Springer
[75] Ledoux, M.; Talagrand, M., Probability in Banach Spaces, Ergeb. Math. (3), vol. 33 (1991), Springer, (paperback 2011) · Zbl 0748.60004
[76] LePage, R. D.; Mandrekar, V., Equivalence-singularity dichotomies from zero-one laws, Proc. Am. Math. Soc., 31, 251-254 (1972) · Zbl 0207.48501
[77] Lifshits, M. A., Gaussian Random Functions, Mathematics and Its Applications, vol. 322 (1995), Kluwer · Zbl 0832.60002
[78] Liu, T. S.; van Rooij, A., Transformation groups and absolutely continuous measures, Indag. Math., 71, 225-231 (1968) · Zbl 0155.46001
[79] Liu, T. S.; van Rooij, A.; Wang, J-K., Transformation groups and absolutely continuous measures II, Indag. Math., 73, 57-61 (1970) · Zbl 0188.45406
[80] Löwner, K., Grundzüge einer Inhaltslehre im Hilbertschen Raume, Ann. Math.. (Bers, Lipman, Charles Loewner: Collected Papers (1988), Birkhäuser), 40, 106-123 (1939), reprinted · JFM 65.1174.02
[81] Ludkovsky, S. V., Properties of quasi-invariant measures on topological groups and associated algebras, Ann. Math. Blaise Pascal, 6, 33-45 (1999) · Zbl 0936.22004
[82] Mackey, G. W., Borel structure in groups and their duals, Trans. Am. Math. Soc., 85, 134-165 (1957) · Zbl 0082.11201
[83] Marcus, M. B.; Rosen, J., Markov Processes, Gaussian Processes, and Local Times, Cambridge Studies in Advanced Mathematics, vol. 100 (2006) · Zbl 1129.60002
[84] McCrudden, M., On the supports of absolutely continuous Gauss measures on connected Lie groups, Monatshefte Math., 98, 295-310 (1984) · Zbl 0544.60018
[85] McCrudden, M., On the supports of Gauss measures on algebraic groups, Math. Proc. Camb. Philos. Soc., 96, 437-445 (1984) · Zbl 0555.60010
[86] McCrudden, M.; Wood, R. M., On the support of absolutely continuous Gauss measures, (Probability Measures on Groups, VII. Probability Measures on Groups, VII, Oberwolfach, 1983. Probability Measures on Groups, VII. Probability Measures on Groups, VII, Oberwolfach, 1983, Lecture Notes in Math., vol. 1064 (1984), Springer), 379-397
[87] Miller, H. I.; Ostaszewski, A. J., Group actions and shift-compactness, J. Math. Anal. Appl., 392, 23-39 (2012) · Zbl 1251.54035
[88] Morris, S. A., Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Soc. Lecture Note Series, vol. 29 (1977), Cambridge University Press · Zbl 0446.22006
[89] Mospan, Y. V., A converse to a theorem of Steinhaus, Real Anal. Exch., 31, 291-294 (2005) · Zbl 1142.28303
[90] von Neumann, J., Die Einführung analytischer Parameter in topologischen Gruppen, Ann. Math., 34, 170-179 (1933), (Collected Works II, 366-386, Pergamon, 1961) · Zbl 0006.30003
[91] J. von Neumann, Review of [80]Math. Reviews; J. von Neumann, Review of [80]Math. Reviews
[92] von Neumann, J., Zum Haarsche Mass in topologischen Gruppen, Compos. Math., 1, 106-114 (1934), (Collected Works II, 445-453, Pergamon, 1961) · Zbl 0008.24602
[93] von Neumann, J., The uniqueness of Haar’s measure, Mat. Sb., 1, 721-734 (1936), (Collected Works IV, 91-104, Pergamon, 1962) · JFM 62.1231.01
[94] Ostaszewski, A. J., Beyond Lebesgue and Baire III: Steinhaus’s theorem and its descendants, Topol. Appl., 160, 1144-1154 (2013) · Zbl 1285.22005
[95] Ostaszewski, A. J., Effros, Baire, Steinhaus and non-separability, M.E. Rudin Memorial Issue. M.E. Rudin Memorial Issue, Topol. Appl., 195, 265-274 (2015) · Zbl 1347.54054
[96] Oxtoby, J. C., Invariant measures in groups which are not locally compact, Trans. Am. Math. Soc., 60, 215-237 (1946) · Zbl 0063.06073
[97] Oxtoby, J. C., Measure and Category, Graduate Texts in Math., vol. 2 (1972), Springer, (1st ed. 1972)
[98] J.C. Oxtoby, A commentary on [50][63][64][67]; J.C. Oxtoby, A commentary on [50][63][64][67]
[99] Pantsulaia, G. R., On an invariant Borel measure in Hilbert space, Bull. Pol. Acad. Sci., Math., 52, 47-51 (2004) · Zbl 1124.28013
[100] Parthasarathy, K. R., Probability Measures on Metric Spaces (2005), Acad. Press: AMS, reprinted: · Zbl 1188.60001
[101] Paterson, A. L.T., Amenability, Math. Surv. Monogr., vol. 29 (1988), Amer. Math. Soc. · Zbl 0648.43001
[102] Pettis, B. J., On continuity and openness of homomorphisms in topological groups, Ann. Math. (2), 52, 293-308 (1950) · Zbl 0037.30501
[103] Piccard, S., Sur les ensembles de distances des ensembles de points d’un espace Euclidien, Mém., vol. 13 (1939), Univ. Neuchâtel, 212 pp · JFM 65.1170.03
[104] Prokaj, V., A characterization of singular measures, Real Anal. Exch., 29, 805-812 (2003/2004) · Zbl 1062.28002
[105] Pugachëv, O. V., Quasi-invariance of Poisson distributions with respect to transformations of configurations, Dokl. Math., 77, 420-423 (2008) · Zbl 1167.60009
[106] Rogers, C. A.; Jayne, J.; Dellacherie, C.; Topsøe, F.; Hoffmann-Jørgensen, J.; Martin, D. A.; Kechris, A. S.; Stone, A. H., Analytic Sets (1980), Academic Press · Zbl 0451.04001
[107] Rudin, W., Fourier Analysis on Groups (1962), Wiley, (reprinted 1990) · Zbl 0107.09603
[108] Rudin, W., Functional Analysis (1991), McGraw-Hill, (1st ed. 1973) · Zbl 0867.46001
[109] Sadasue, G., On absolute continuity of the Gibbs measure under translations, J. Math. Kyoto Univ., 41, 257-276 (2001) · Zbl 1011.60017
[110] Sadasue, G., On quasi-invariance of infinite product measures, J. Theor. Probab., 21, 3, 571-585 (2008) · Zbl 1149.22008
[111] Schwartz, L., Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Tata Institute of Fundamental Research Studies in Mathematics, vol. 6 (1973), Oxford University Press · Zbl 0298.28001
[112] Shepp, L. A., Distinguishing a sequence of random variables from a translate of itself, Ann. Math. Stat., 36, 1107-1112 (1965) · Zbl 0136.40108
[113] Shimomura, H., Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations, J. Funct. Anal., 187, 406-441 (2001) · Zbl 0997.58004
[114] Simmons, S. M., A converse Steinhaus theorem for locally compact groups, Proc. Am. Math. Soc., 49, 383-386 (1975) · Zbl 0318.43002
[115] Skorohod, A. V., Integration in Hilbert Space, Ergebnisse Math., vol. 79 (1974), Springer
[116] Smoleński, W., On quasi-invariance of product measures, Demonstr. Math., 11, 3, 801-805 (1978) · Zbl 0394.28002
[117] Solecki, S., Amenability, free subgroups, and Haar null sets in non-locally compact groups, Proc. Lond. Math. Soc. (3), 93, 693-722 (2006) · Zbl 1115.22002
[118] Steinhaus, H., Sur les distances des points de mesure positive, Fundam. Math., 1, 83-104 (1920) · JFM 47.0179.02
[119] Stroock, D. W., Probability Theory. An Analytic View (2011), Cambridge University Press, (1st ed. 1993) · Zbl 1223.60001
[120] Tarieladze, V., Characteristic functionals of probabilistic measures in DS-groups and related topics, J. Math. Sci., 211, 137-296 (2015) · Zbl 1358.60007
[121] F. Topsøe, J. Hoffmann-Jørgensen, Analytic spaces and their application, in: [106]; F. Topsøe, J. Hoffmann-Jørgensen, Analytic spaces and their application, in: [106]
[122] Weil, A., L’intégration dans les groupes topologiques, Actualités Scientifiques et Industrielles, vol. 1145 (1965), Hermann, (1st ed. 1940)
[123] Xia, D. X., Measure and Integration Theory on Infinite-Dimensional Spaces. Abstract Harmonic Analysis, Pure Appl. Math., vol. 48 (1972), Academic Press
[124] Yamasaki, Y., Translationally invariant measure on the infinite-dimensional vector space, Publ. Res. Inst. Math. Sci., 16, 693-720 (1980) · Zbl 0475.28007
[125] Yamasaki, Y., Measures on Infinite-Dimensional Spaces (1985), World Scientific · Zbl 0591.28012
[126] Yosida, K., Functional Analysis, Classics in Mathematics (1995), Springer, Reprint of the sixth (1980) edition · Zbl 0152.32102
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.