Abstract
It is shown that a measurable function from an atomless Loeb probability space (Ω, A, P) to a Polish space is at least continuum-to-one valued almost everywhere. It follows that there is no injective mapping h : [0, 1] → Ω such that h([a, b]) is Loeb measurable for each 0 ≤ a ≤ b ≤ 1 and P(h([0,1])) > 0. Thus, when an atomless Loeb measurable algebra on an internal set of cardinality continuum is imposed on the unit interval [0, 1] through a bijection, it cannot contain the Borel algebra.
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Keisler, H.J., Sun, Y. (2001). Loeb Measures and Borel Algebras. In: Schuster, P., Berger, U., Osswald, H. (eds) Reuniting the Antipodes — Constructive and Nonstandard Views of the Continuum. Synthese Library, vol 306. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9757-9_10
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DOI: https://doi.org/10.1007/978-94-015-9757-9_10
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