Uniform logical proofs for Riesz representation theorem, Daniell-Stone theorem and Stone's representation theorem for probability algebras

Abstract

Riesz representation theorem, Daniell-Stone theorem for Daniell integrals and Stone's representation theorem for probability and measure algebras are three important classical results in analysis concerning existence of measures with certain properties. Many proofs of these theorems can be found in the literature of analysis, from elementary ones which use ordinary techniques from measure theory, to more sophisticated ones, such as those employing techniques from nonstandard analysis, in particular for Riesz representation theorem. In this paper, as the first goal, we give new proofs for all these three theorems. Our proofs have a mild logical flavor and are uniform in the sense that they are all based on the same general idea and rely on the application of the same technical tool from logic to measure theory, namely logical compactness theorem. In fact, as the second goal of the paper, we try to reveal more the power of logical methods in analysis in particular measure theory, and make stronger connections between analysis and logic. We use the setting of "integration logic" which is a logical framework (and one of the forms of probability logics) for studying measure and probability structures by logical means. Indeed, we elaborate this setting and use its expressive power and a version of compactness theorem holding in it to show its application in measure theory by giving new proofs for the above-mentioned measure existence theorems. As mentioned, an advantage of these proofs is that they are all given in a uniform way since they are all based on the logical compactness theorem. The paper is mostly written for general mathematicians, in particular the people active in analysis or logic as the main audience. So it is self-contained and the reader does not need to have any advanced prerequisite knowledge from logic or measure theory.