What theory are they?
Set theory comes to mind but is that too broad?
Your question is somewhat vague. However, measure theory starts with the notion of collections of subsets, of a measurable set, which are closed under countable unions and countable intersections. A measure is then defined as a non-negative function of these subsets, which must follow certain rules. Essentially it has to be countably additive.
Measure theory is a fairly advanced subject, and it falls under the heading of real analysis. You need to be comfortable with basic set theory (as in all math), limits, sequences, series, continuity, convergence, and other topics that you would see in a year long sequence in undergraduate real analysis. Do you have a more specific question?
Here are two good books to start with if you are starting to learn the subject.
https://www.amazon.com/Lebesgue-Int...sr_1_1?ie=UTF8&s=books&qid=1238983053&sr=8-1" by Alan Weir
https://www.amazon.com/Lebesgue-Int...sr_1_1?ie=UTF8&s=books&qid=1238983133&sr=1-1" by Frank Jones
Separate names with a comma.