Compactness of Loeb Spaces

In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In §1 we prove that Loeb spaces are compact under various assumptions, and in §2 we prove that Loeb spaces are not compact under various other assumptions. The results in §1 and §2 give a quite complete answer to a question of D. Ross in [R1], [R2] and [R3]. 0. Introduction In [R1] and [R2] D. Ross asked: Are (bounded) Loeb measure spaces compact? J. Aldaz then, in [A], constructed a counterexample. But Aldaz’s example is atomic, while most of Loeb measure spaces people are interested are atomless. So Ross reasked his question in [R3]: Are atomless Loeb measure spaces compact? In this paper we answer the question. Let’s assume that all measure spaces mentioned throughout this paper are atomless probability spaces. Given a probability space (Ω,Σ, P ). A subfamily C ⊆ Σ is called compact if for any D ⊆ C, D has f.i.p. i.e. finite intersection property, implies ⋂ D 6= ∅. We call a compact family C inner-regular on Ω if for any A ∈ Σ P (A) = sup{P (C) : C ⊆ A ∧ C ∈ C}. A probability space (Ω,Σ, P ) is called compact if Σ contains an inner-regular compact subfamily. Clearly, the definition of compactness is a generalization of Radon spaces with no topology involved. In fact, Ross proved in [R2] that a compact probability space is essentially Radon, i.e. one can topologize the space so that every measurable set A contains a compact subset of measure at least half of the measure of A. Loeb measure spaces are important tools in nonstandard analysis (see, for example, [AFHL] and [SB]). Ross proved in [R2] that every compact probability space is the Mathematics Subject Classification Primary 28E05, 03H05, 03E35. The research of the first author was supported by NSF postdoctoral fellowship #DMS-9508887. This research was started when the first author spent a wonderful year as a visiting assistant professor in University of Illinois-Urbana Champaign during 94-95. He is grateful to the logicians there. The research of the second author was supported by The Israel Science Foundation administered by The Israel Academy of Sciences and Humanities. This paper is number 613 on the second author’s publication list. 1