The Study And Applications Of Loeb Measure In Nontopological Sets

In1975, Peter Loeb found Loeb measure and utilized them to represent the standard measures via the standard part map. Since then, this technique has turned out tobe very useful, specially in applications to probability theory. However,which measurescan be represented by Loeb measures? Anderson proved that if X is a Hausdorf spaceand μ a Borel measure, then L(μ) st1()=μ if and only if μ is Radon, and alsothat μ can be replaced by an internal discrete measure with hyperfnite support. Afterthen, other scholars found a way to represent more general measures,namely those thatare 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 space. The outline of this thesis is as follows:In Section1, the start, development and current research status of nonstandardanalysis are presented. In Section2, frst some fundamental theories of nonstandardanalysis are gived, and then several diferent kinds of nonstandard models are discussed,and properties of them are also obtained.In Section3, Loeb measure space (X, L(A), L(μ)) has been constructed by twokinds of diferent methods in internal measure space (X, A, μ),and their consistencyhas been proved.In Section4, The concepts of monad and standard part map are presented intopological space, then, in the Hausdorf topological space, and under fnite Borelmeasure, The defnition of the regular,Radon, τ Smooth of measure μ is gived, Andlast the relevant theorems and proofs are studied. In Section5, In the nontopological sets, lattice L is defned, and then regularand τ Smooth of measure μ with respect to L are expressed by Loeb outer measure,Further, the corresponding theorems and corollary are proved. It is proved that everyfnite, fnitely additive measure that is τ Smooth and outer regular with respect tosome lattice L has a τ Smooth extension to a σ algebra containing all union of setsfrom L.