Loeb Measures and Borel Algebras
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چکیده
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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تاریخ انتشار 2005