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Perfect Measures and Maps

This paper is designed to provide a solid introduction to the theory of nonmeasurable calculus. In this context basic concepts are reviewed and fundamental facts are presented. Perfect measures and maps are shown to of a vital importance to support the theory, and therefore these concepts are treated separately as well. In particular, many equivalent statements to the property of being perfect are given. The last part of the paper deals with a setting arising in the investigation on extension of measures in the case of perturbations of their -algebras.