The terms "measure," "measurable," etc. have very precise technical definitions (usually involving sigma-algebras) that can make them appear difficult to understand. However, the technical nature of the definitions is extremely important, since it gives a firm footing to concepts that are the basis for much of analysis (including some of the slippery underpinnings of calculus).

For example, every definition of an integral is based on a particular measure: the Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. The study of measures and their application to integration is known as measure theory.

A measure is defined as a nonnegative real function from a delta-ring such that

(1)

where is the empty set, and

(2)

for any finite or countable collection of pairwise disjoint sets in such that is also in .

If is -finite and is bounded, then can be extended uniquely to a measure defined on the -algebra generated by .

is said to be a probability measure on a set if and is a -algebra.

In the usual definition of a probability measure (or, more precisely a nontrivial -additive measure), a measure is a real-valued function on the Power Set of an infinite set that satisfies the following properties:

1. and ,

2. If then ,

3. for all (nontriviality),

4. If , are pairwise disjoint, then

(3)

(Jech 1997).

A measure may be extended by completion. The subsets of sets with measure zero form a -ring . By "changing" sets in on a set from , a -ring which is the completion of with respect to is obtained.

The measure is called complete if . If is not complete, it may be extended to by setting , where and .