Measure (mathematics) - Definition

Definition

Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:

• Non-negativity:

• Null empty set:

• Countable additivity (or σ-additivity): For all countable collections of pairwise disjoint sets in Σ:

One may require that at least one set E has finite measure. Then the null set automatically has measure zero because of countable additivity, because, so .

If only the second and third conditions of the definition of measure above are met, and μ takes on at most one of the values ±∞, then μ is called a signed measure.

The pair is called a measurable space, the members of are called measurable sets. If is another measurable space then a function is called measurable if for every Y-measurable set, the inverse image is X-measurable i.e. . The composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the set of measurable functions as the arrows.

A triple (X, Σ, μ) is called a measure space. A probability measure is a measure with total measure one (i.e., μ(X) = 1); a probability space is a measure space with a probability measure.

For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other sources. For more details see Radon measure.