Borel partitions of unity and lower Carathéodory multifunctions

Srivastava, S.

Srivastava, S.

Fundamenta Mathematicae, Tome 146 (1995), p. 239-249 / Harvested from The Polish Digital Mathematics Library

Résumé

We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in $A (ℰ \otimes ℬ (X))$ into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential equations and inclusions and in mathematical economics. As a key tool we prove that if A is an analytic subset of E × X and if $U_{n} : n \in w$ is a sequence of Borel sets in A such that $A = \cup_{n} U_{n}$ and the section $U_{n} (e)$ is open in A(e), e ∈ E, n ∈ w, then there exist Borel functions $p_{n} : A \to [0, 1]$ , n ∈ w, such that for every e ∈ E, $p_{n} (e, \cdot) : n \in w$ is a locally Lipschitz partition of unity subordinate to $U_{n} (e) : n \in w$ .

Publié le : 1995-01-01

Zbl 0821.04001

EUDML-ID : urn:eudml:doc:212064

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     author = {S. Srivastava},
     title = {Borel partitions of unity and lower Carath\'eodory multifunctions},
     journal = {Fundamenta Mathematicae},
     volume = {146},
     year = {1995},
     pages = {239-249},
     zbl = {0821.04001},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv146i3p239bwm}
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Srivastava, S. Borel partitions of unity and lower Carathéodory multifunctions. Fundamenta Mathematicae, Tome 146 (1995) pp. 239-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv146i3p239bwm/

Bibliographie

[00000] [AC] J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984.

[00001] [AF] J.-P. Aubin and H. Frankowska, Set Valued Analysis, Systems and Control: Found. Appl., Birkhäuser, Boston, 1990.

[00002] [F] A. Fryszkowski, Carathéodory type selectors of set-valued maps of two variables, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 41-46. | Zbl 0358.54002

[00003] [J] J. Janus, A remark on Carathéodory type selections, Le Matematiche 25 (1986), 3-13. | Zbl 0678.28005

[00004] [Kuc] A. Kucia, On the existence of Carathéodory selectors, Bull. Polish Acad. Sci. Math. 32 (1984), 233-241. | Zbl 0562.28004

[00005] [Kur] K. Kuratowski, Topology, Vol. I, PWN, Warszawa, and Academic Press, New York, 1966.

[00006] [KPY] T. Kim, K. Prikry and N. C. Yannelis, Carathéodory-type selections and random fixed point theorems, J. Math. Anal. Appl. 122 (1987), 393-407. | Zbl 0629.28007

[00007] [Łoj] S. Łojasiewicz, Jr., Parametrizations of convex sets, J. Approx. Theory, to appear.

[00008] [Lou] A. Louveau, A separation theorem for $Σ_{1}^{1}$ sets, Trans. Amer. Math. Soc. 260 (1980), 363-378. | Zbl 0455.03021

[00009] [Mic] E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 363-382.

[00010] [Mil] D. E. Miller, Borel selectors for separated quotients, Pacific J. Math. 91 (1980), 187-198. | Zbl 0477.54008

[00011] [Mo] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980.

[00012] [MR] A. Maitra and B. V. Rao, Generalizations of Castaing's theorem on selectors, Colloq. Math. 42 (1979), 295-300. | Zbl 0427.28010

[00013] [BR] B. V. Rao and K. P. S. Bhaskara Rao, Borel spaces, Dissertationes Math. (Rozprawy Mat.) 190 (1981).

[00014] [Ri] B. Ricceri, Selections of multifunctions of two variables, Rocky Mountain J. Math. 14 (1984), 503-517. | Zbl 0552.54010

[00015] [Ry] L. Rybiński, On Carathéodory type selections, Fund. Math. 125 (1985), 187-193. | Zbl 0614.28005

[00016] [S1] S. M. Srivastava, A representation theorem for closed valued multifunctions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 511-514. | Zbl 0446.54015

[00017] [S2] S. M. Srivastava, Approximations and approximate selections of upper Carathéodory multifunctions, Boll. Un. Mat. Ital. A (7) 8 (1994), 251-262. | Zbl 0813.54012

[00018] [SS1] H. Sarbadhikari and S. M. Srivastava, Random theorems in topology, Fund. Math. 136 (1990), 65-72. | Zbl 0728.28009

[00019] [SS2] H. Sarbadhikari and S. M. Srivastava, Random versions of extension theorems of Dugundji type and fixed point theorems, Boll. Un. Mat. Ital., to appear. | Zbl 0794.54049

[00020] [Y] N. C. Yannelis, Equilibria in non-cooperative models of computations, J. Econom. Theory 41 (1987), 96-111.