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Measure Theory
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Tell me everything I need to know about measure theory.
>>
>>7787723
Measures exist.
>>
A few things to look up

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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