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Riesz representation theorem, Daniell-Stone theorem for Daniell integrals andStone's representation theorem for probability and measure algebras are threeimportant classical results in analysis concerning existence of measures withcertain properties. Many proofs of these theorems can be found in theliterature of analysis, from elementary ones which use ordinary techniques frommeasure theory, to more sophisticated ones, such as those employing techniquesfrom nonstandard analysis, in particular for Riesz representation theorem. Inthis 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 theyare all based on the same general idea and rely on the application of the sametechnical tool from logic to measure theory, namely logical compactnesstheorem. In fact, as the second goal of the paper, we try to reveal more thepower of logical methods in analysis in particular measure theory, and makestronger connections between analysis and logic. We use the setting of"integration logic" which is a logical framework (and one of the forms ofprobability logics) for studying measure and probability structures by logicalmeans. Indeed, we elaborate this setting and use its expressive power and aversion of compactness theorem holding in it to show its application in measuretheory 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 auniform way since they are all based on the logical compactness theorem. Thepaper is mostly written for general mathematicians, in particular the peopleactive in analysis or logic as the main audience. So it is self-contained andthe reader does not need to have any advanced prerequisite knowledge from logicor measure theory.

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