In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if

Motivation

The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by We now wish to construct some two-dimensional Lebesgue measure \lambda^2 on the plane \R^2 as a product measure. Naively, we would take the 𝜎-algebra on \R^2 to be the smallest 𝜎-algebra containing all measurable "rectangles" for While this approach does define a measure space, it has a flaw. Since every singleton set has one-dimensional Lebesgue measure zero, for subset A of \R. However, suppose that A is a non-measurable subset of the real line, such as the Vitali set. Then the \lambda^2-measure of is not defined but and this larger set does have \lambda^2-measure zero. So this "two-dimensional Lebesgue measure" as just defined is not complete, and some kind of completion procedure is required.

Construction of a complete measure

Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ0, μ0) of this measure space that is complete. The smallest such extension (i.e. the smallest σ-algebra Σ0) is called the completion of the measure space. The completion can be constructed as follows: Then (X, Σ0, μ0) is a complete measure space, and is the completion of (X, Σ, μ). In the above construction it can be shown that every member of Σ0 is of the form A ∪ B for some A ∈ Σ and some B ∈ Z, and

Examples

Properties

Maharam's theorem states that every complete measure space is decomposable into measures on continua, and a finite or countable counting measure.