A **measure space** is a fundamental concept in measure theory, which is a branch of mathematics that deals with the study of size, length, area, and volume in a rigorous way. A measure space provides a framework for quantifying the "size" of sets, particularly in the context of integration and probability theory.
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A measure space is a set , a sigma-algebra of its subsets, and a measure defined on . Specifying only leaves both the measurable questions and their weights undetermined.
For a concrete probability space, take , its power set, and weights . Define by adding the weights of the elements in . ThenA probability measure is exactly a measure normalized by , usually written . An indicator function has expected value equal to its event's probability. For a different continuous example, with the restricted Lebesgue measure is a probability space; the event has probability .