Star-finite representations of measure spaces

Abstract

In nonstandard analysis, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">^{\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-finite sets are infinite sets which nonetheless possess the formal properties of finite sets. They permit a synthesis of continuous and discrete theories in many areas of mathematics, including probability theory, functional analysis, and mathematical economics. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">^{\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-finite models are particularly useful in building new models of economic or probabilistic processes. It is natural to ask what standard models can be obtained from these <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">^{\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-finite models. In this paper, we show that a rich class of measure spaces, including the Radon spaces, are measure-preserving images of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">^{\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-finite measure spaces, using a construction introduced by Peter A. Loeb [<bold>15</bold>]. Moreover, we show that a number of measure-theoretic constructs, including integrals and conditional expectations, are naturally expressed in these models. It follows that standard models which can be expressed in terms of these measure spaces and constructs can be obtained from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">^{\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-finite models.