рус eng
«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2023. Vol 44

On Radon Barycenters of Measures on Spaces of Measures

Author(s)
Vladimir I. Bogachev1,2,3,4, Svetlana N. Popova2,5

1Lomonosov Moscow State University, Moscow, Russian Federation

2National Research University Higher School of Economics, Moscow, Russian Federation

3Saint Tikhon’s Orthodox University, Moscow, Russian Federation

4Moscow Center for Fundamental and Applied Mathematics, Moscow, Russian Federation

5Moscow Institute of Physics and Technology (State University), Dolgoprudny, Russian Federation

Abstract
We study metrizability of compact sets in spaces of Radon measures with the weak topology. It is shown that if all compacta in a given completely regular topological space are metrizable, then every uniformly tight compact set in the space of Radon measures on this space is also metrizable. It is proved that the property that compact sets of measures on a given space are metrizable is preserved for products of this space with spaces that can be embedded into separable metric spaces. In addition, we construct a Radon probability measure on the space of Radon probability measures on a completely regular space such that its barycenter is not a Radon measure.
About the Authors

Vladimir I. Bogachev, Dr. Sci. (Phys.–Math.), Prof., Moscow State University, Moscow, 119991, Russian Federation, vibogach@mail.ru

Svetlana N. Popova, Cand. Sci. (Phys.Math.), Junior Researcher, Moscow Institute of Physics and Technology, Dolgoprudny, 141701, Russian Federation popovaclaire@mail.ru

For citation
Bogachev V. I., Popova S. N. On Radon Barycenters of Measures on Spaces of Measures. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 44, pp. 19–30. https://doi.org/10.26516/1997-7670.2023.44.19
Keywords
Radon measure, barycenter, metrizable compact set of measures
UDC
517.9
MSC
28C15, 28A33, 46E27
DOI
https://doi.org/10.26516/1997-7670.2023.44.19
References
  1. Acciaio B., Beiglb¨ock M., Pammer G. Weak transport for non-convex costs and model-independence in a fixed-income market. Math. Finance, 2021, vol. 31, no. 4. pp. 1423–1453. https://doi.org/10.1111/mafi.12328
  2. Alibert J.-J., Bouchitt´e G., Champion T. A new class of costs for optimal transport planning. European J. Appl. Math., 2019, vol. 30, no. 6. pp. 1229–1263. https://doi.org/10.1017/S0956792518000669
  3. Arkhangel’ski˘ı A.V., Ponomarev V.I. Fundamentals of General Topology. Problems and Exercises. Translated from the Russian. Reidel, Dordrecht, 1984, 415 p.
  4. Backhoff-Veraguas J., Beiglb¨ock M., Pammer G. Existence, duality, and cyclical monotonicity for weak transport costs. Calc. Var. Partial Differ. Equ., 2019, vol. 58, no. 203, pp. 1–28. https://doi.org/10.1007/s00526-019-1624-y
  5. Backhoff-Veraguas J. Pammer G. Stability of martingale optimal transport and weak optimal transport. Ann. Appl. Probab., 2022, vol. 32, no. 1. pp. 721–752. https://doi.org/10.1214/21-AAP1694
  6. Bogachev V.I. Measure Theory. In 2 vols. Berlin, Springer, 2007, vol. 1, 500 p.; vol. 2, 575 p. https://doi.org/10.1007/978-3-540-34514-5
  7. Bogachev V.I. Weak Convergence of Measures. Amer. Math. Soc., Providence, Rhode Island, 2018, 286 p. https://doi.org/10.1090/surv/234
  8. Bogachev V.I. Kantorovich problems with a parameter and density constraints. Siberian Math. J., 2022, vol. 63, no. 1. pp. 34–47. https://doi.org/10.1134/S0037446622010037
  9. Bogachev V.I. The Kantorovich problem of optimal transportation of measures: new directions of research. Uspehi Matem. Nauk (Russian Math. Surveys), 2022, vol. 77, no. 5. pp. 3–52. https://doi.org/10.4213/rm10074 (in Russian)
  10. Bogachev V.I., Malofeev I.I. Nonlinear Kantorovich problems depending on a parameter. The Bulletin of Irkutsk State University. Series Mathematics., 2022, vol. 41. pp. 96–106. https://doi.org/10.26516/1997-7670.2022.41.96
  11. Bogachev V.I., Rezbaev A.V. Existence of solutions to the nonlinear Kantorovich problem of optimal transportation. Mathematical Notes, 2022, vol. 112, no. 3. pp. 369–377. https://doi.org/10.1134/S0001434622090048
  12. Engelking P. General Topology. Warszawa, Polish Sci. Publ., 1977, 626 p.
  13. Gozlan N., Roberto C., Samson P.-M., Tetali P. Kantorovich duality for general transport costs and applications. J. Funct. Anal., 2017, vol. 273, no. 11. pp. 3327–3405. https://doi.org/10.1016/j.jfa.2017.08.015
  14. Steen L., Seebach J. Counterexamples in Topology. 2nd ed. New York, Springer, 1978, 244 p.
  15. Topsøe F. Preservation of weak convergence under mappings. Ann. Math. Statist., 1967, vol. 38, no. 6. pp. 1661–1665. https://doi.org/10.1214/aoms/1177698600

Full text (english)