# Borel partitions of unity and lower Carathéodory multifunctions

Fundamenta Mathematicae (1995)

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

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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 -

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