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 some two-dimensional Lebesgue measure λ2 on the plane R2 as a product measure. Naïvely, we would take the σ-algebra on R2 to be B ⊗ B, the smallest σ-algebra containing all measurable "rectangles" A1 × A2 for Ai ∈ B.
While this approach does define a measure space, it has a flaw. Since every singleton set has one-dimensional Lebesgue measure zero,
for "any" subset A of R. However, suppose that A is a non-measurable subset of the real line, such as the Vitali set. Then the λ2-measure of {0} × A is not defined, but
and this larger set does have λ2-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 a measure on the continuum, and a finite or countable counting measure.