Loeb测度在非拓扑集上研究及其应用.pdf

Loeb Æ Æ Æ 1975 Peter Loeb Loeb ℄ Loeb Anderson X Hausdorff µ ∗ −1 Borel L( µ) ◦ st () = µ µ Radon ∗ µ. Loeb ℄ τ − κ− κ card(T ), Æ ℄ (X, A , µ) Loeb (X, L(A ), L(µ)), Hausdorff Borel µ Radon τ − L, Loeb ℄ L τ − L τ − ℄ τ − L σ Æ: Loeb ℄ τ − The Study and Applications of Loeb Measure in Nontopological sets Specialty: Applied Mathematics Name: Zheng Huanping Instructor: Chen Dongli ABSTRACT In 1975, Peter Loeb found Loeb measure and utilized them to represent the stan dard measures via the standard part map. Since then, this technique has turned out to be very useful, specially in applications to probability theory. However,which measures can be represented by Loeb measures? Anderson proved that if X is a Hausdorff space ∗ −1 and µ a Borel measure, then L( µ) ◦ st () = µ if and only if µ is Radon, and also ∗ that µ can be replaced by an internal discrete measure with hyperfinite support. After then, other scholars found a way to represent more general measures,namely those that are regular and τ −smooth, by using the outer measure generated by a Loeb measure. In this thesis, we shall work with a κ−saturated nonstandard model, κ card(T ), extend these results to abstract measure sp