Definition of a sigma-algebra, of a measure, of a measurable function. Some constructions (product measures, push-forward, Caratheodory) Properties of the Lebesgue measure (regularity, behaviour under diffeomorphism) Construction of the integral relative to a measure Lp spaces Riesz representation theorems Radon-Nikodym theorem
depends what you need it for. if you only want to know the gist of it to bring it up in your autistic name-droppings, you have to realize that it is a way to formalize the whole aspect of "defining length", and from there, the fields of probability, integration, analysis fall out. For example, you can learn a good amount of Markov Chains without knowing measure and freely using Fubini to justify interchange of integral and summation(for integrable functions).
its a very dry field and you start from "basics"(sigma algebra, borel sets), and work from the ground up. so you could follow what >>7787761 said.
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