Some results about Borel sets in descriptive set theory of hyperfinite sets

Živaljević, Boško

doi:10.2307/2274650

Cited by 8 publications

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“…In [2], the question of when two internal sets A, B are bijective by a countably determined function is answered: precisely when |^4|/|5| is neither infinitesimal nor infinite. In [5], this result is generalized to arbitrary nonvanishing sets. We shall resolve the question for Borel sets A, B.…”

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confidence: 98%

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Vanishing Borel sets

Schilling¹

1998

J. symb. log.

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Henson and Ross [1] answered the question of when two hyperfinite sets A, B in an ℵ1-saturated nonstandard universe are bijective by a Borel function: precisely when ∣A∣/∣B∣ ≈ 1. Živaljević [5] generalized this result to nonvanishing Borel sets. He defined a set to be nonvanishing if it is Loeb-measurable and has finite, non-zero measure with respect to some Loeb counting measure. He then showed that two nonvanishing Borel sets are Borel bijective just in case they have the same finite, non-zero measure with respect to some Loeb counting measure.Here we shall complete the cycle, for Borel sets at least. For N ∈ *N, let λN be the internal counting measure given by λN(A) = ∣A∣/N for A internal. Then for vanishing Loeb-measurable sets B, it is natural to consider the Dedekind cut (BL, BR) on *N consisting of those N for which B has 0λN-measure infinity and zero, respectively. We show that, for all vanishing Borel sets B, B and BL are Borel bijective. It follows that vanishing Borel sets B and C are Borel bijective if, and only if, BL = CL. Combined with Živaljević's result, we can characterize when arbitrary Borel sets are Borel bijective: precisely when they have the same measure with respect to all Loeb counting measures.In the final section, we generalize in a similar way results of [2] and [5] to characterize when two Borel sets are bijective by a countably determined function: precisely when, for all N, one has 0λN-measure 0 if and only if the other also has 0λN-measure 0.

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confidence: 98%

“…[5]). Let A, B be nonvanishing (measurable) sets, andlet M, N G & be such that °XM{A) and°k^{B) are both nonzero and finite; then A, B are bijective by a countably determined function if and only if M/N is neither infinitesimal nor infinite.…”

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confidence: 99%

Vanishing Borel sets

Schilling¹

1998

J. symb. log.

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“…It would be interesting to give a sufficient and necessary condition for two true (i.e., non-Borel) nj subsets P and Q of a hyperfinite set X that ensure the existence of a nj bijection between them. For Borel P and Q it was shown in [8] that P and Q are Borel bijective if and only if P and Q have the same nonvanishing L(ß) measure for some counting, uniformly distributed internal measure ß. (Here, the word nonvanishing means that the measure of P and Q are not 0 or oc.…”

Section: Resultsmentioning

confidence: 99%

Π¹₁ functions are almost internal

Živaljević¹

1995

Trans. Amer. Math. Soc.

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Abstract.In Analytic mappings on hyperfinite sets [Proc. Amer. Math. Soc. (1993), 587-596] Henson andRoss asked for what hyperfinite sets S and T does there exists a bijection f from 5 onto T whose graph is a projective subset of S x r? In particular, when is there a nj bijection from S onto 77 In this paper we prove that given an internal, bounded measure p, any nj function is L(p) a.e. equal to an internal function, where L(p) is the Loeb measure associated with p. It follows that if two nj subsets S and T of a hyperfinite set X are nj bijective, then S and T have the same measure for every uniformly distributed counting measure p.. When 5 and T are internal it turns out that any nj bijection between them must already be Borel. We also prove that if a nj graph in the product of two hyperfinite sets X and Y is universal for all internal subsets of Y , then \X\ > 2lyl , which is a partial answer to Henson and Ross's Problem 1.5. At the end we prove some standard results about the projections and a structure of co-proper if-analytic subsets of the product of two completely regular HausdorfF topological spaces with open vertical sections. We were able to prove the above results by revealing the structure of n! subsets of the products X x Y of two internal sets X and y, all of whose F-sections are Z?(íc) sets.

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“…Therefore, D is defined by the conjunction of formula (3) and (4) nEco Conversely, if 21Y[ ~ IX ] or 2 IYI ~ IX t then, as was shown by Henson and Ross in [5] (see also [12]), there exists a Borel onto map defined on X and with values in the internal power set of Y. Then, by a~l-saturation, /'(x) is N~ nonempty and not equal to Y if and only if it is nonempty and E ~ over the family A~(z)(n C w).…”

Section: Corollary 26 Let X and Y Be Internal Sets And Let P C_ X Xmentioning

confidence: 94%

Graphs with ∏ 1 0 (K)Y-sections

Živaljević

1993

Arch Math Logic

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We prove that a Borel subset of the product of two internal sets X and Y all of whose Y-sections are II~176 sets is the intersection (union) of a countable sequence of Borel graphs with internal Y-sections. As a consequence we prove some standard results about the domains of graphs in the product of two topological spaces all of whose horizontal section are compact (open) sets. A version of classical Vitali-Lusin theorem for those types of graphs is given as well as a new proof (and an extension) of a classical result of Kunugui. IntroductionThe investigation of structure of graphs in the product X • Y of two internal sets X and Y with Y-sections of fixed Borel class (in the Descriptive Set Theory of Hyperfinite Sets) has been inspired by a curious result of the authors of [7] about the structure of Borel functions. They proved that any function whose graph is a Borel subset of the product X • Y can be covered by countably many internal functions (provided that the nonstandard universe is cvl-saturated ). This in turn shows that any Borel function can be expressed as a countable union of restrictions of internal functions to disjoint Borel sets in X. In [13] this result has been extended to arbitrary Borel graphs all of whose Y-sections are internal sets. Earlier, the Loeb measure preserving property of injective Borel functions has been noticed by the authors of [5], who in fact proved that any Borel function possesses an internal lifting (in our terminology).

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Some results about Borel sets in descriptive set theory of hyperfinite sets

Cited by 8 publications

References 3 publications

Vanishing Borel sets

Vanishing Borel sets

Π¹₁ functions are almost internal

Graphs with ∏ 1 0 (K)Y-sections

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