Since your question is formulated somewhat provocatively, let me answer in a similar spirit: The only measure theory you need to know is that the space of Radon measures is the topological dual of the space of continuous functions. And since you're not interested in details and lemmas, you're already done.
Put less facetiously, what you need from measure theory is mainly the Riesz representation theorem in sufficiently general form. This is already treated in textbooks on abstract analysis (as part of integration theory), and you can avoid books on (modern) abstract measure theory (which evolved, in part, as a foundation for probability theory). A list of suitable texts can be found in Measure theory treatment geared toward the Riesz representation theorem, to which I would add Conway's recent Course in Abstract Analysis, AMS 2012.
(If you're interested in even "cooler" right-hand sides, you really need to make your question more specific.)