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, (X, Σ, μ) is complete if and only if
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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 (R, B, λ). We now wish to construct twodimensional Lebesgue measure λ^{2} on the plane R^{2} as a product measure. Naïvely, we would take the σalgebra on R^{2} to be B ⊗ B, the smallest σalgebra containing all measurable "rectangles" A_{1} × A_{2} for A_{i} ∈ B.
While this approach does define a measure space, it has a flaw. Since every singleton set has onedimensional Lebesgue measure zero,
for "any" subset A of R. However, suppose that A is a nonmeasurable subset of the real line, such as the Vitali set. Then the λ^{2}measure of {0} × A is not defined, but
and this larger set does have λ^{2}measure zero. So, "twodimensional 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, Σ_{0}, μ_{0}) of this measure space that is complete. The smallest such extension (i.e. the smallest σalgebra Σ_{0}) is called the completion of the measure space.
The completion can be constructed as follows:
 let Z be the set of all subsets of μmeasure zero subsets of X (intuitively, those elements of Z that are not already in Σ are the ones preventing completeness from holding true);
 let Σ_{0} be the σalgebra generated by Σ and Z (i.e. the smallest σalgebra that contains every element of Σ and of Z);
 there is a unique extension μ_{0} of μ to Σ_{0} given by the infimum
Then (X, Σ_{0}, μ_{0}) 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 Σ_{0} is of the form A ∪ B for some A ∈ Σ and some B ∈ Z, and
Examples
 Borel measure as defined on the Borel σalgebra generated by the open intervals of the real line is not complete, and so the above completion procedure must be used to define the complete Lebesgue measure.
 ndimensional Lebesgue measure is the completion of the nfold product of the onedimensional Lebesgue space with itself. It is also the completion of the Borel measure, as in the onedimensional case.
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