Representation of measures via the standard part map
In this dissertation we study the representation of standard measures and contents via Loeb measures and the standard part map, extending previously known results to a nontopological setting. In addition, a characterization of the completely regular and Hausdorff spaces which are universally Loeb measurable is given. Applications of these techniques include a new nonstandard proof of the Riesz Representation Theorem, as well as the construction of the projective hull, an object akin to the projective limit of a projective system of measure spaces. Loeb's functional approach to nonstandard measure theory is also considered, and a simplification of his work is proposed.
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign