Abstract
Ideas and techniques from standard and nonstandard theories of measure spaces and Banach spaces are brought together to give a new approach to the study of the extension of vector measures. Applications of our results lead to simple new proofs for theorems of classical measure theory. The novelty lies in the use of the principle of extension by continuity (for which we give a nonstandard proof) to obtain in an unified way some notable theorems which have been obtained by Fox, Brooks, Ohba, Diestel, and others. The methods of proof are quite different from those used by previous authors, and most of them are realized by means of nonstandard analysis.
Highlights
Let Ω be a nonempty fixed set and ] a real-valued positive measure on a ring R of subsets of Ω; the measure is assumed t]oo(f⋃bde∞ nis=cj1ooEiunnnt)tma=bel∑ym∞ nab=de1dr]si(tEiovfne)Ri.nAtafhnuednsdeifanm⋃see∞ ntn=h1taaEtl npifri(osEbanlle)smois a sequence in R, in measure theory is that of finding conditions under which a countably additive measure on a ring R can be extended to a countably additive measure on a wider class of sets containing R
In this work we study the extension of vector valued set functions in the framework of nonstandard analysis
In the semimetric space (R, ρ) the set operations are continuous [5]
Summary
Let Ω be a nonempty fixed set and ] a real-valued positive measure on a ring R of subsets of Ω; the measure is assumed t]oo(f⋃bde∞ nis=cj1ooEiunnnt)tma=bel∑ym∞ nab=de1dr]si(tEiovfne)Ri.nAtafhnuednsdeifanm⋃see∞ ntn=h1taaEtl npifri(osEbanlle)smois a sequence in R, in measure theory is that of finding conditions under which a countably additive measure on a ring R can be extended to a countably additive measure on a wider class of sets containing R. R. So, if ] is a bounded set function on a ring R with values in a Banach space, what are the possibilities to obtain an extension?. This result was obtained for the first time by Fox [3]. The central construction in this approach is the notion of nonstandard hull introduced by Luxemburg [21] This notion is a useful tool in studying vector measures and Banach spaces, and a construction arising naturally throughout nonstandard analysis. We obtain a result concerning the concentration of s-bounded vector measures on a specific set from the nonstandard extension of R.
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