Weakly metrically transitive measures and nonmeasurable sets

Abstract

In the previous chapters we were concerned with various extensions of ??-finite measures, which are obtained by using certain types of nonmeasurable sets or nonmeasurable functions. It was also mentioned that sometimes absolutely nonmeasurable sets can occur for concrete classes of measures (e.g., Vitali sets for the class of all translation-invariant extensions of the Lebesgue measure λ=λ₁ on the real line R = R₁). Obviously, the latter sets turn out to be useless from the point of view of the measure extension problem.