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Pushing down Loeb measures. (English) Zbl 0782.28006
We develop new methods of representing measures via Loeb measures. The main idea is to use an extension theorem of R. Sikorski in order to construct certain $$\sigma$$-homomorphisms from the $$\sigma$$-algebra generated by a system of subsets into the Loeb $$\sigma$$-algebra. Since the standard part map can be seen as a special $$\sigma$$-homomorphism from the Borel $$\sigma$$-algebra into the Loeb $$\sigma$$-algebra we obtain rather general results about the representations of measures with nonstandard techniques. Several applications are given. Finally, we extend some results to additive internal set functions defined on a lattice of sets and solve positively a problem of J. Aldaz.

##### MSC:
 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 28E05 Nonstandard measure theory
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