Measure theory: transplantation theorems for inner premeasures

Abstract

Abstract.The main result is a new transplantation theorem for the inner premeasures of the author, with a few related theorems. These results have basic implications for example for the construction of Radon measures. They received a certain inspiration from the treatment of Radon measures in the treatise of Fremlin on measure theory. Mathematics Subject Classification (1991). 28A12, 28C15.Keywords. Inner premeasures and their maximal inner extensions, Transplantation theorems, Radon premeasures and Radon measures.1. Introduction. The present article is part of the author's new systematization in measure and integration initiated in [3], of which the latest accounts are [8,10]. As before we concentrate on the inner version. We recall that its basic concepts are the inner • premeasures and their maximal inner • extensions(• = στ with = finite, σ = sequential, τ = nonsequential), and that its basic devices are the inner • envelopes. We shall often make free use of the concepts and results set up so far.One of the final chapters in [3] was devoted to transplantation theorems for inner premeasures. The central results were for premeasures and hence of a certain simple character: but these results often appear in combination with topological compactness or with set-theoretic • = στ compactness, and thus lead to consequences for Radon measures and in the new • = στ theories. We recall the former main theorem [3, 18.10] = [6, 2.3], which also explains the word transplantation. Let X denote a nonvoid set.