# Borel partitions of unity and lower Carathéodory multifunctions

Fundamenta Mathematicae (1995)

- Volume: 146, Issue: 3, page 239-249
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topSrivastava, S.. "Borel partitions of unity and lower Carathéodory multifunctions." Fundamenta Mathematicae 146.3 (1995): 239-249. <http://eudml.org/doc/212064>.

@article{Srivastava1995,

abstract = {We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in $A(ℰ ⊗ ℬ(X))$ 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 $\{U_n : n ∈ w\}$ is a sequence of Borel sets in A such that $A=∪_n U_n$ and the section $U_n(e)$ is open in A(e), e ∈ E, n ∈ w, then there exist Borel functions $p_n : A → [0,1]$, n ∈ w, such that for every e ∈ E, $\{p_n(e,·) : n ∈ w\}$ is a locally Lipschitz partition of unity subordinate to $\{U_n(e) : n ∈ w\}$.},

author = {Srivastava, S.},

journal = {Fundamenta Mathematicae},

keywords = {Carathéodory functions and multifunctions; Carathéodory selections; fixed points; lower Carathéodory multifunction; separable Banach space; Borel -field; Polish space; extensions of continuous functions; Borel sets; Borel functions; locally Lipschitz partition of unity},

language = {eng},

number = {3},

pages = {239-249},

title = {Borel partitions of unity and lower Carathéodory multifunctions},

url = {http://eudml.org/doc/212064},

volume = {146},

year = {1995},

}

TY - JOUR

AU - Srivastava, S.

TI - Borel partitions of unity and lower Carathéodory multifunctions

JO - Fundamenta Mathematicae

PY - 1995

VL - 146

IS - 3

SP - 239

EP - 249

AB - We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in $A(ℰ ⊗ ℬ(X))$ 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 ${U_n : n ∈ w}$ is a sequence of Borel sets in A such that $A=∪_n U_n$ and the section $U_n(e)$ is open in A(e), e ∈ E, n ∈ w, then there exist Borel functions $p_n : A → [0,1]$, n ∈ w, such that for every e ∈ E, ${p_n(e,·) : n ∈ w}$ is a locally Lipschitz partition of unity subordinate to ${U_n(e) : n ∈ w}$.

LA - eng

KW - Carathéodory functions and multifunctions; Carathéodory selections; fixed points; lower Carathéodory multifunction; separable Banach space; Borel -field; Polish space; extensions of continuous functions; Borel sets; Borel functions; locally Lipschitz partition of unity

UR - http://eudml.org/doc/212064

ER -

## References

top- [AC] J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984.
- [AF] J.-P. Aubin and H. Frankowska, Set Valued Analysis, Systems and Control: Found. Appl., Birkhäuser, Boston, 1990.
- [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. Zbl0358.54002
- [J] J. Janus, A remark on Carathéodory type selections, Le Matematiche 25 (1986), 3-13. Zbl0678.28005
- [Kuc] A. Kucia, On the existence of Carathéodory selectors, Bull. Polish Acad. Sci. Math. 32 (1984), 233-241. Zbl0562.28004
- [Kur] K. Kuratowski, Topology, Vol. I, PWN, Warszawa, and Academic Press, New York, 1966.
- [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. Zbl0629.28007
- [Łoj] S. Łojasiewicz, Jr., Parametrizations of convex sets, J. Approx. Theory, to appear.
- [Lou] A. Louveau, A separation theorem for ${\Sigma}_{1}^{1}$ sets, Trans. Amer. Math. Soc. 260 (1980), 363-378. Zbl0455.03021
- [Mic] E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 363-382.
- [Mil] D. E. Miller, Borel selectors for separated quotients, Pacific J. Math. 91 (1980), 187-198. Zbl0477.54008
- [Mo] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980.
- [MR] A. Maitra and B. V. Rao, Generalizations of Castaing's theorem on selectors, Colloq. Math. 42 (1979), 295-300. Zbl0427.28010
- [BR] B. V. Rao and K. P. S. Bhaskara Rao, Borel spaces, Dissertationes Math. (Rozprawy Mat.) 190 (1981).
- [Ri] B. Ricceri, Selections of multifunctions of two variables, Rocky Mountain J. Math. 14 (1984), 503-517. Zbl0552.54010
- [Ry] L. Rybiński, On Carathéodory type selections, Fund. Math. 125 (1985), 187-193. Zbl0614.28005
- [S1] S. M. Srivastava, A representation theorem for closed valued multifunctions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 511-514. Zbl0446.54015
- [S2] S. M. Srivastava, Approximations and approximate selections of upper Carathéodory multifunctions, Boll. Un. Mat. Ital. A (7) 8 (1994), 251-262. Zbl0813.54012
- [SS1] H. Sarbadhikari and S. M. Srivastava, Random theorems in topology, Fund. Math. 136 (1990), 65-72. Zbl0728.28009
- [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. Zbl0794.54049
- [Y] N. C. Yannelis, Equilibria in non-cooperative models of computations, J. Econom. Theory 41 (1987), 96-111.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.