Integration is a linear functional on a vector space of functions. The RRT identifies the dual of continuous functions on a compact Hausdorff space with countably additive Borel measures. A very simple result is the dual of bounded functions on any set is the space of finitely additive measures. (E.g., https://keithalewis.github.io/tandon/vs.html) D. J. H. Garling has a very clever proof of the theorem based on this that is "short" if you know about the Stone-Cech compactification, the Hahn-Banach theorem, and the Carathéodory extension theorem. :-) https://www.cambridge.org/core/journals/mathematical-proceed...