“School of Mathematics”
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| Paper IPM / M / 16935 |
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| Abstract: | |
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Integration logic is a logical (model theoretic) framework for studying measure and probability structures by logical means. The Daniell�??Stone theorem for Daniell integrals and Riesz representation theorem are two important classical results in analysis concerning existence of measures with certain properties. Many proofs of the Riesz representation theorem can be found in the literature from elementary ones which use ordinary techniques from measure theory and analysis, to more sophisticated ones that highly employ techniques from other fields of mathematics such as nonstandard analysis. There are also a few proofs for the Daniell�??Stone theorem. This paper pursues two main goals. One is to give novel and uniform proofs for both these theorems using some ideas from logic. The second and maybe even more important one is to try to reveal more the power of the logical methods in mathematics in particular measure theory, and make stronger connections between analysis and logic. Our proofs are uniform in the sense that they are based on the same general idea and rely on the application of the same technical tool from logic to measure theory, namely the logical compactness theorem. We use the setting of ïntegration logic" mentioned above, elaborate it and use its expressive power and a version of the compactness theorem holding in it to prove our results. The paper is mostly written for general mathematicians, in particular the people active in logic or analysis as the main audience. It is self-contained and the reader does not need to have any advanced prerequisite knowledge from logic or measure theory.
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