The celebrated Radon-Nikodym theorem and Stone representation theorem for measure algebras are two important classical results in analysis. This paper pursues two main goals. One is to give new proofs for these theorems by using ideas from logic and application of an important theorem, namely, the logical compactness theorem. The second and even more important goal is to try to reveal more the power of logical methods in analysis in particular measure theory, and make stronger connections between two fields of analysis and logic. Through the paper, we use a logical setting called ”integration logic” which is a framework for studying measure and probability structures through logical means. The paper is mostly written for general mathematicians, in particular the people active in logic or analysis as the main audiences. It is self-contained and does not require advanced prerequisite knowledge from logic or analysis.