Selection theorem in L¹

Andrzej Nowak; Celina Rom

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2006)

  • Volume: 26, Issue: 1, page 123-127
  • ISSN: 1509-9407

Abstract

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Let F be a multifunction from a metric space X into L¹, and B a subset of X. We give sufficient conditions for the existence of a measurable selector of F which is continuous at every point of B. Among other assumptions, we require the decomposability of F(x) for x ∈ B.

How to cite

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Andrzej Nowak, and Celina Rom. "Selection theorem in L¹." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 26.1 (2006): 123-127. <http://eudml.org/doc/271151>.

@article{AndrzejNowak2006,
abstract = {Let F be a multifunction from a metric space X into L¹, and B a subset of X. We give sufficient conditions for the existence of a measurable selector of F which is continuous at every point of B. Among other assumptions, we require the decomposability of F(x) for x ∈ B.},
author = {Andrzej Nowak, Celina Rom},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {multifunction; measurable selector; continuous selector; decomposable set},
language = {eng},
number = {1},
pages = {123-127},
title = {Selection theorem in L¹},
url = {http://eudml.org/doc/271151},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Andrzej Nowak
AU - Celina Rom
TI - Selection theorem in L¹
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2006
VL - 26
IS - 1
SP - 123
EP - 127
AB - Let F be a multifunction from a metric space X into L¹, and B a subset of X. We give sufficient conditions for the existence of a measurable selector of F which is continuous at every point of B. Among other assumptions, we require the decomposability of F(x) for x ∈ B.
LA - eng
KW - multifunction; measurable selector; continuous selector; decomposable set
UR - http://eudml.org/doc/271151
ER -

References

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  1. [1] S.M. Ageev and D. Repovs, On selection theorems with decomposable values, Topol. Methods Nonlinear Anal. 15 (2000), 385-399. Zbl0971.54017
  2. [2] A.V. Arutyunov, Special selectors of multivalued mappings (in Russian), Dokl. Akad. Nauk Ross. Akad. Nauk 377 (3) (2001), 298-300. English translation: Dokl. Math. 63 (2) (2001), 182-184. Zbl1048.54011
  3. [3] A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 69-86. Zbl0677.54013
  4. [4] A. Fryszkowski, Continuous selections for a class of non-convex multivalued maps, Studia Math. 76 (1983), 163-174. Zbl0534.28003
  5. [5] F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149-182. Zbl0368.60006
  6. [6] C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72. Zbl0296.28003
  7. [7] S.J. Leese, Multifunctions of Souslin type, Bull. Austral. Math. Soc. 11 (1974), 395-411. Zbl0287.04005
  8. [8] A. Nowak and C. Rom, Decomposable hulls of multifunctions, Discuss. Math. Differ. Incl. Control Optim. 22 (2002), 233-241. Zbl1032.26025
  9. [9] Cz. Olech, Decomposability as a substitute for convexity, Multifunctions and Integrands: Stochastic Analysis, Approximation and Optimization, Proc. Conf. Catania, Italy, June 7-16, 1983 (G. Salinetti, ed.); Lecture Notes in Math., vol. 1091, Springer-Verlag, Berlin, 1984, pp. 193-205. 
  10. [10] A.A. Tolstonogov and D.A. Tolstonogov, Lₚ-continuous extreme selectors of multifunctions with decomposable values: Existence theorems, Set-Valued Anal. 4 (1996), 173-203. Zbl0847.54019
  11. [11] D.H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (5) (1977), 859-903. Zbl0407.28006

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