Nonstandard measure theory: avoiding pathological sets

Abstract

The main results in this paper concern representing Lebesgue measure by nonstandard measures which avoid certain pathological sets. An (external) set E is S-thin if InfmA|A standard,* A ⊇ E A\, \supseteq \,E = 0 and Q-thin if Inf*mA|A internal, A ⊇ E A\, \supseteq \,E = 0. It is shown that any *finite sample which represents Lebesgue measure avoids every S-thin set and that given any Q-thin set E there is a *finite sample avoiding E which represents Lebesgue measure. In the last part of the paper a particular pathological set H ⊆ ∗ [ 0 , 1 ] {\mathcal {H}}\,\, \subseteq \, * \left [ {0,\,1} \right ] is constructed which is important for the study of approximate limits, derivatives etc. It is shown that every *finite sample which represents Lebesgue measure assigns inner measure zero and outer measure one to this set and that Loeb measure does the same. Finally, it is shown that Loeb measure can be extended to a σ \sigma -algebra including H {\mathcal {H}} in such a way that H {\mathcal {H}} is assigned zero measure.