Rigorous branch of analysis that we can use to build up the theory of probability distributions and continuous random variables.
Quick intuition: a set of points with measure 0 occupies no volume in the space we are measuring (i.e., a line in ). Positive measure means it does occupy some volume (i.e., a filled polygon).
An individual point has measure 0, as does the union of countably many sets. This means, for example, that the set of rationals has measure 0.