Application of transfinite numbers and infinitesimals to measure theory

摘要: Measure theory provides one of the most inviting areas in which transfinite and infinitesimal numbers non-standard analysis may be applied. This is so because their use becomes not just a convenient tool but an essential requirement for generalization theory. In Chapter 1 we set-theoretic approach to establish basis our subsequent work. The process involves injective maps (monomorphisms) allows us contrast technique with that using more concrete less direct ultrapower method. chapter sufficient framework allow selfcontained examination basic properties extended real line carried out 2. Non-standard measure developed 3 where construct premeasure F it define y as extension Lebesgue all sets on line. constructed point such its standard part agrees latter defined. It finitely additive sense thus natural solution "easy problem measure" solved first by Banach. 4 show are measurable apply some well known subsets R find approximate measures them. We also obtain cardinality results taking parts those cases set under consideration measurable.