A measurable space is a set together with a collection of subsets of which is a sigma algebra.

The elements of are called measurable sets.

A measurable space is the correct object on which to define a measure; will be the collection of sets which actually have a measure. We normally want to ensure that contains all the sets we will ever want to use. We usually cannot take to be the collection of all subsets of because the axiom of choice often allows one to construct sets that would lead to a contradiction if we gave them a measure (even zero). For the real numbers, Vitali's theorem states that cannot be the collection of all subsets if we hope to have a measure that returns the length of an open interval.