Skip to main content
SpringerLink
Log in

Survey of measurable selection theorems: An update

  • Measurable Selections
  • Conference paper
  • First Online:
Measure Theory Oberwolfach 1979

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 794))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

New Bibliography

  1. N. U. Ahmed and K. L. Teo, Comments on a selector theorem in Banach spaces, J. Opt. Th. Appl., 19 (1976), 117–118.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Ascherl and J. Lehn, Two principles for extending probability measures, Manuscripta Math. 21 (1977), 43–50.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. W. Baggett and A. Ramsey, A functional analytic proof of a selection lemma, preprint.

    Google Scholar 

  4. J. A. Bain, Optimal measurable selections for conditional expectations, preprint.

    Google Scholar 

  5. J-M. Bismut, Intégrales convexes et probabilités, J. Math. Anal. Appl., 42 (1973), 639–673.

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Bourbaki, Topologie Général, Hermann, 1958.

    Google Scholar 

  7. J. Bourgain, Meetbare beschrijring van Baire Verzamelingen (Measurable description of Baire sets), Thesis, Vrije Universiteit Brussel, 1977.

    Google Scholar 

  8. _____, A stabilization property and its applications in the theory of sections, Séminaire Choquet (Initiation à l'Analyse), 1977/78, no. 5.

    Google Scholar 

  9. J. P. Burgess, A selector principle for Σ1/1 equivalence relations, Michigan Math. J., 24 (1977), 65–76.

    Article  MathSciNet  MATH  Google Scholar 

  10. _____, A measurable selection theorem, preprint.

    Google Scholar 

  11. _____, A selection theorem for group actions, Pacific J. Math., 80 (1979), 333–336.

    Article  MathSciNet  MATH  Google Scholar 

  12. _____, Two selection theorems, Bull. Math. Soc. Grèce, 18 (1977), 121–136.

    MathSciNet  MATH  Google Scholar 

  13. _____, Sélections mesurables pour relations d'équivalences analytiques à classes Gδ, preprint.

    Google Scholar 

  14. _____, Classical hierarchies from a modern standpoint, Part I. C-sets, preprint.

    Google Scholar 

  15. _____, Careful choices, preprint.

    Google Scholar 

  16. J. P. Burgess and D. E. Miller, Remarks on invariant descriptive set theory, Fund. Math., 90 (1975), 53–75.

    MathSciNet  MATH  Google Scholar 

  17. C. Castaing, À propos de l'existence des sections séparément mesurables et séparément continues d'une multiapplication séparément mesurable et séparément semi-continue inférieurement, Séminaire d'Analyse Convexe, Montpellier (1976), Exposé no. 6.

    Google Scholar 

  18. A. Cellina, A selection theorem, Rend. Sem. Mat. Univ. Padova, 55 (1976), 143–149.

    MathSciNet  MATH  Google Scholar 

  19. G. Choquet, Theory of capacities, Annales de l'Institut Fourier 5 (1953/54), 131–295.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Cornea and G. Licea, Une démonstration unifiée des théorèmes de sections de P. A. Meyer, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 10 (1968), 198–202.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Costé, Contribution à la théorie de l'intégration multivoque, Thèse, l'Université Pierre et Marie Curie, Paris 6, 1977.

    Google Scholar 

  22. G. Debs, Sélections maximales d'une multi-application, preprint.

    Google Scholar 

  23. ____, Sélection d'une multi-application à valeurs Gβ, preprint.

    Google Scholar 

  24. C. Dellacherie, Capacités et Processus Stochastiques, Springer, 1972.

    Google Scholar 

  25. _____, Ensembles analytiques: théorèmes de séparation et applications, Sém. Probabilité IX, 1973/74, 336–372, Lecture Notes in Math., Vol. 465, Springer, 1975.

    Google Scholar 

  26. _____, Un cours sur les ensembles analytiques, London Symposium on Analytic Sets, 1979.

    Google Scholar 

  27. C. Dellacherie and P. A. Meyer, Un nouveau théorème de projection et de section, Sém. Probabilité IX, 1973/74, 239–245, Lecture Notes in Math., Vol. 465, Springer, 1975.

    MathSciNet  MATH  Google Scholar 

  28. E. E. Doberkat, On a representation of measurable automaton-transformations by stochastic automata, J. Math. Anal. Appl., 69 (1979), 455–468.

    Article  MathSciNet  MATH  Google Scholar 

  29. _____, Deterministic representations of stochastic and nondeterministic automata, preprint.

    Google Scholar 

  30. S. Dolecki, Extremal measurable selections, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 25 (1977), 355–360.

    MathSciNet  MATH  Google Scholar 

  31. J. L. Doob, Stochastic Processes, Wiley, 1953.

    Google Scholar 

  32. G. A. Edgar, Measurable weak sections, Illinois J. Math., 20 (1976), 630–646.

    MathSciNet  MATH  Google Scholar 

  33. E. G. Effros, Transformation groups and C*-algebras, Ann. of Math., 81 (1965), 38–55.

    Article  MathSciNet  MATH  Google Scholar 

  34. I. Ekeland, Review of [CV3], Bull. Amer. Math. Soc., 84 (1978), 950–957.

    Article  MathSciNet  Google Scholar 

  35. M. P. Ershov (Yershov), Extensions of measures and stochastic equations, Theory Probability Appl., 19 (1974), 431–444.

    Article  Google Scholar 

  36. _____, The Choquet theorem and stochastic equations, Analysis Mathematica, 1 (1975), 259–271.

    Article  MathSciNet  MATH  Google Scholar 

  37. _____, On selectors in abstract spaces, Uspekhi, Mat. Nauk, 33 (1978), 205–206 (Russian).

    MathSciNet  MATH  Google Scholar 

  38. _____, Some selection theorems for partitions of sets without topology, these Proceedings.

    Google Scholar 

  39. I. V. Evstigneev, Measurable choice and the continuum axiom, Soviet Math. Dokl., 19 (1978), 1–5.

    MATH  Google Scholar 

  40. S. Graf, Measurable weak selections, these Proceedings.

    Google Scholar 

  41. _____, A measurable selection theorem for compact-valued maps, Manuscripta Math. 27 (1979), 341–352.

    Article  MathSciNet  MATH  Google Scholar 

  42. _____, Induced σ-homomorphisms and a parametrization of measurable sections via extremal preimage measures, preprint.

    Google Scholar 

  43. O. Hajek, Discontinuous differential equations I, preprint.

    Google Scholar 

  44. M. Hasumi, A continuous selection theorem for extremally disconnected spaces, Math. Ann., 179 (1969), 83–89.

    Article  MathSciNet  MATH  Google Scholar 

  45. H. Hermes, On continuous and measurable selections and the existence of solutions of generalized differential equations, Proc. Amer. Math. Soc., 29 (1971), 535–542.

    Article  MathSciNet  MATH  Google Scholar 

  46. C. J. Himmelberg, T. Parthasarathy, and F. S. Van Vleck, On measurable relations, Fund. Math., to appear.

    Google Scholar 

  47. C. J. Himmelberg and F. S. Van Vleck, Corrigendum and addendum to "Multifunctions on abstract measurable spaces and application to stochastic decision theory," Ann. Mat. Pura. Appl., to appear.

    Google Scholar 

  48. _____, Existence of solutions for generalized differential equations with unbounded right-hand side, preprint.

    Google Scholar 

  49. A. D. Ioffe, Survey of measurable selection theorems: Russian literature supplement, SIAM J. Control and Opt., 16 (1978), 728–732.

    Article  MathSciNet  MATH  Google Scholar 

  50. _____, Single-valued representations of set-valued mappings, Trans. Amer. Math. Soc., 252 (1979), 133–145.

    Article  MathSciNet  MATH  Google Scholar 

  51. _____, One-to-one Carathéodory representation theorem for multifunctions with uncountable values, Fund. Math., to appear.

    Google Scholar 

  52. _____, Representation theorems for measurable multifunctions, these Proceedings.

    Google Scholar 

  53. A. Kozek and Z. Suchanecki, Conditional expectations of selectors and Jensen's inequality, Proc. Conference on Mathematical Statistics, Wisła, 1978.

    Google Scholar 

  54. _____, Multifunctions of faces for conditional expectations of selectors and Jensen's inequality, Institute of Mathematics, Polish Academy of Sciences, Preprint 154, 1978.

    Google Scholar 

  55. Le Van Tu, Omega-Polish spaces and measurable selections, J. Austral. Math. Soc., Ser. A 23 (1977), 257–265.

    Article  MathSciNet  MATH  Google Scholar 

  56. V. L. Levin, On Borel selections of many-valued mappings, Siberian Math. J., 19 (1978), 617–623 (in Russian).

    Article  MATH  Google Scholar 

  57. _____, Review of [CV3], Math. Rev. 7169, 57 (1978), 941.

    Google Scholar 

  58. V. Losert, A measure space without the strong lifting property, Math. Ann., 239 (1979), 119–128.

    Article  MathSciNet  MATH  Google Scholar 

  59. _____, A counterexample on measurable selections and strong lifting, preprint.

    Google Scholar 

  60. G. Mägerl, Zu den Schnittsätzen von Michael und Kuratowski & Ryll-Nardzewski, Dissertation, Universität Erlangen-Nürnberg, 1977.

    Google Scholar 

  61. _____, A unified approach to measurable and continuous selections, Trans. Amer. Math. Soc., 245 (1978), 443–452.

    Article  MathSciNet  MATH  Google Scholar 

  62. A. Maitra, Analytic sets with countable sections, Fund. Math., to appear.

    Google Scholar 

  63. _____, Extensions of measurable functions, preprint.

    Google Scholar 

  64. A. Maitra and B. V. Rao, Generalizations of Castaing's theorems on selectors, Colloq. Math., to appear.

    Google Scholar 

  65. A. Maitra, B. V. Rao, and K. P. S. Bhaskara Rao, A problem in extension of measures, Illinois J. Math., 23 (1979), 211–216.

    MathSciNet  MATH  Google Scholar 

  66. D. A. Martin, Borel determinancy, Ann. of Math., 102 (1975), 363–371.

    Article  MathSciNet  Google Scholar 

  67. R. D. Mauldin, The unboundedness of the Cantor-Bendixson order of some analytic sets, Pacific J. Math., 74 (1978), 167–177.

    Article  MathSciNet  MATH  Google Scholar 

  68. _____, Borel parametrizations, Trans. Amer. Math. Soc., 250 (1979), 223–234.

    Article  MathSciNet  MATH  Google Scholar 

  69. _____, Some selection theorems and problems, these Proceedings.

    Google Scholar 

  70. P. A. Meyer, Une présentation de la théorie des ensembles sousliniens. Application aux processes stochastiques, Séminaire de Théorie de Potential. Institut Henri Poincairé, 7e année, 201–218.

    Google Scholar 

  71. _____, Une nouvelle démonstration des théorèmes de section, Sém. Probabilité III, 1967/68, pp. 155–159. Lecture Notes in Math., Vol. 88, Springer, 1969.

    Google Scholar 

  72. D. E. Miller, On the measurability of orbits in Borel actions, Proc. Amer. Math. Soc., 63 (1977), 165–170.

    Article  MathSciNet  MATH  Google Scholar 

  73. _____, A selector for equivalence relations with Gδ orbits, Proc. Amer. Math. Soc., 72 (1978), 365–369.

    MathSciNet  MATH  Google Scholar 

  74. _____, Borel selectors for separated quotients, preprint.

    Google Scholar 

  75. J. A. Mühlbauer, Bayes-Losungen für Sequenztests mit kontinuierlicher Zeit und nicht notwendig endlich vielen Hypothesen, Inaugural-Dissertation, Marburg/Lahn (1977).

    Google Scholar 

  76. S. Nababan, A Fillipov-type lemma for functions involving delays and its application to time-delayed optimal control problems, J. Opt. Th. and Appl., 27 (1979), 357–376.

    Article  MathSciNet  MATH  Google Scholar 

  77. T. Nishiura, Two counterexamples for measurable relations, Rocky Mountain J. Math., 9 (1979), 499–501.

    Article  MathSciNet  MATH  Google Scholar 

  78. A. Nowak, Discounted dynamic programming on Euclidean spaces, Applicationes Mathematicae 16 (1979), 465–473.

    MathSciNet  MATH  Google Scholar 

  79. Phan Van Chuong, Densité des sections extremales d'une multiapplication mesurable, Séminaire d'Analyse Convexe, Montpellier, 1979, Exposé No. 5.

    Google Scholar 

  80. U. Rieder, Measurable selection theorems for optimization problems, Manuscripta Math., 24 (1978), 115–131.

    Article  MathSciNet  MATH  Google Scholar 

  81. _____, Measurable selection of fixed points, preprint.

    Google Scholar 

  82. J. Saint-Pierre, Caractérisations des F-sousliniens d'un produit, C.R. Acad. Sci. Paris Sér. A, 283 (1976), 583–585.

    MathSciNet  MATH  Google Scholar 

  83. _____, Séparation simultanée d'un ensemble analytique de suites d'ensembles F-Sousliniens, Ibid., 285 (1977), 933–936.

    MathSciNet  MATH  Google Scholar 

  84. _____, Caractérisation des F-sousliniens d'un produit et théorèmes de sections, Séminaire D'Analyse Convexe, Montpellier (1977), Exposé no. 8.

    Google Scholar 

  85. _____, Séparation d'un schema de Souslin, Ibid., Exposé no. 16.

    Google Scholar 

  86. _____, Points exposés d'une multi-application à valeurs convexes de Rn, Ibid, 1978, Exposé no. 7.

    Google Scholar 

  87. J. Saint-Raymond, Boréliens à coupes Kσ, Bull. Soc. Math. France, 104 (1976), 389–400.

    MathSciNet  MATH  Google Scholar 

  88. _____, Fonctions boréliennes sur un quotient, Bull. Sci. Math. 2e Sér., 100 (1976), 1–7.

    MathSciNet  MATH  Google Scholar 

  89. _____, Sélections mesurables pour la structure borélienne d'Effros, Bull. Sci. Math., 2e Sér., 101 (1977), 353–355.

    MathSciNet  MATH  Google Scholar 

  90. G. Salinetti and R. J-B. Wets, On the convergence of closed-valued measurable multifunctions, preprint.

    Google Scholar 

  91. H. Sarbadhikari, A selection theorem for multifunctions, Proc. Amer. Math. Soc., 71 (1978), 285–288.

    Article  MathSciNet  MATH  Google Scholar 

  92. _____, Selection theorems for partitions of complete metric spaces, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys., to appear.

    Google Scholar 

  93. H. Sarbadhikari and S. M. Srivastava, Parametrization of Gδ-valued multifunctions, Trans. Amer. Math. Soc., to appear.

    Google Scholar 

  94. L. Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Oxford University Press, 1973.

    Google Scholar 

  95. S. Shreve and D. Bertsekas, Equivalent stochastic and deterministic optimal control problems, Proc. IEEE Conf. on Decision and Control, 1976, 705–709.

    Google Scholar 

  96. _____, Universally measurable policies in dynamic programming, Mathematics of Operations Research, 4 (1979), 15–30.

    Article  MathSciNet  MATH  Google Scholar 

  97. S. M. Srivastava, A representation theorem for closed valued multifunctions, Bull. Pol. Acad. Sci. Sér. Sci. Math. Astronom. Phys., to appear.

    Google Scholar 

  98. _____, A representation theorem for Gδ-valued multifunctions, Amer. J. Math., to appear.

    Google Scholar 

  99. _____, Selection theorems for Gδ-valued multifunctions, Trans. Amer. Math. Soc., 254 (1979), 283–293.

    MathSciNet  MATH  Google Scholar 

  100. _____, Studies in the theory of measurable multifunctions, Thesis, Indian Stat. Inst. (1978).

    Google Scholar 

  101. V. V. Srivatsa, Existence of measurable selectors and parametrizations for Gδ-valued multifunctions, preprint.

    Google Scholar 

  102. M. Talagrand, Sélection mesurable de mesures maximales simpliciales, Bull. Sci. Math., 2e Sér. 102 (1978), 49–56.

    MathSciNet  MATH  Google Scholar 

  103. _____, Non-existence de certaines sections mesurables et application à la théorie du relèvement, C. R. Acad. Sci. Paris Sér. A, 286 (1978), 1183–1185.

    MathSciNet  MATH  Google Scholar 

  104. _____, En général il n'existe pas de relèvement linéaire borélien fort, Ibid,, 287 (1978), 633–634.

    MathSciNet  MATH  Google Scholar 

  105. _____, Non-existence de certaines sections mesurables et contre-exemples en théorie du relèvement, these Proceedings.

    Google Scholar 

  106. M. Valadier, Sur le plongement d'un champ mesurable d'espaces metriques dans un champ trivial, Ann. Inst. Henri Poincaré, Sect. B. to appear; also in Séminaire d'Analyse Convexe, Montpellier (1977), Exposé 14.

    Google Scholar 

  107. M. P. Yershov — see Ershov.

    Google Scholar 

  108. D. H. Wagner, Survey of measurable selection theorems, SIAM J. Control and Opt., 15 (1977), 859–903.

    Article  MathSciNet  MATH  Google Scholar 

  109. J. Warga, An implicit function theorem without differentiability, Proc. Amer. Math. Soc., 69 (1978), 65–69.

    Article  MathSciNet  MATH  Google Scholar 

  110. H. von Weizsäcker, Some negative results in the theory of lifting, Measure Theory (Proc. Conf. Oberwolfach, 1975), pp. 159–172. Lecture Notes in Math., Vol. 541, Springer, 1976.

    Google Scholar 

  111. E. Wesley, On the existence of absolutely measurable selection functions, preprint.

    Google Scholar 

Update of Previous Bibliography

  1. C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer, 1977.

    Google Scholar 

  2. D. Cenzer and R. D. Mauldin, Inductive definitions: Measure and category, Advances in Math., to appear.

    Google Scholar 

  3. _____, Measurable parametrizations and selections, Trans. Amer. Math. Soc., 245 (1978), 399–408.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. J. Himmelberg, T. Parthasarathy, and F. S. Van Vleck, Optimal plans for dynamic programming problems, Mathematics of Operations Research, 1 (1976), 390–394.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. R. Kallman and R. D. Mauldin, A cross section theorem and application to C*-algebras, Proc. Amer. Math. Soc., 69 (1978), 57–61.

    MathSciNet  MATH  Google Scholar 

  6. J. Kaniewski, A selection theorem for partitions of Borel sets into compact subsets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys., 25 (1977), 1073–1081.

    MathSciNet  MATH  Google Scholar 

  7. S. J. Leese, Measurable selections and the uniformisation of Souslin sets, Amer. J. Math., 100 (1978), 19–41.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Lehn, Remark on measurable graph theorems, Proc. Amer. Math. Soc., 63 (1977), 46–48.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Lehn and G. Mägerl, On the uniqueness of preimage measures, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33 (1977), 333–337.

    Article  MATH  Google Scholar 

  10. H. Sarbadhikari, Some uniformization results, Fund. Math., 97 (1977), 209–214.

    MathSciNet  MATH  Google Scholar 

  11. M. Valadier, Convex integrands on Souslin locally convex spaces, Pac. J. Math., 59 (1975), 267–276.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 19301, Paoli, Pennsylvania, U.S.A.

    Daniel H. Wagner (Associates)

Authors
  1. Daniel H. Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Dietrich Kölzow

Rights and permissions

Reprints and permissions

Copyright information

© 1980 Springer-Verlag

About this paper

Cite this paper

Wagner, D.H. (1980). Survey of measurable selection theorems: An update. In: Kölzow, D. (eds) Measure Theory Oberwolfach 1979. Lecture Notes in Mathematics, vol 794. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0088224

Download citation

  • DOI: https://doi.org/10.1007/BFb0088224

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09979-6

  • Online ISBN: 978-3-540-39221-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation