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• W3144056556 abstract "A version of Rado's theorem about the existence of a family of subsets of a given family of sets combinatorially similar to another family of sets is proved for the Loeb space of an internal «-finitely additive measure space. As a corollary we obtain the Loeb measured case of the result of B. Bollobas and N. Th. Varopoulos about the existence of a family of mutually disjoint measurable subsets of the given family of measurable sets, having the prescribed measure. The aim of this paper is to prove a version of R. Rado's theorem for the Loeb space of an internal »-finitely additive measure space. In 1937 R. Rado proved (see [Ra]) an extension of P. Hall's theorem (see [Ha]) about the existence of distinct representatives of a finite family of finite sets. He replaced of elements of a by a general measure function and instead of selecting single elements from each member of a family he selected subsets combinatorially similar to a given family of sets (see the definition below). Although Rado's theorem deals with a very general measure function taking values in a general linearly ordered set, it seems to be restricted by the assumption of finiteness of the family of sets in question. Here, using the finite form of Rado's theorem, we extend Rado's theorem to an infinite number of measurable sets for the class of Loeb spaces of internal *-finitely additive measure spaces. We assume that the reader is familiar with basic facts of nonstandard analysis. For the necessary definitions and elementary properties of a nonstandard universe refer to [StBa] or [HuLo]. Let Jf = (Q, Z, s) be an internal, nonatomic, »-finitely additive measure space, that is, Q is an internal set, I a *-algebra of internal subsets of Q, and s a »-finitely additive measure on X. Nonatomic means that for every element A G I with s(A) not being infinitesimal, there exists a set AsZ with B C A Received by the editors May 26, 1989 and, in revised form, May 31, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 03H05, 28A99; Secondary 05C99. The research was supported by the National Science Foundation and the Research Board of the University of Illinois. ©1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page" @default.
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• W3144056556 date "2010-01-01" @default.
• W3144056556 modified "2023-09-26" @default.
• W3144056556 title "RADO'S THEOREM FOR THE LOEB SPACE OF AN INTERNAL »-FINITELY ADDITIVE MEASURE SPACE" @default.
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