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Concept
Complete measure
Summary
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
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 on the plane as a product measure. Naively, we would take the sigma-algebra on to be the smallest sigma-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 of However, suppose that is a non-measurable subset of the real line, such as the Vitali set. Then the -measure of is not defined but
and this larger set does have -measure zero. So this "two-dimensional Lebesgue measure" as just defined is not complete, and some kind of completion procedure is required.
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:
let Z be the set of all the subsets of the zero-μ-measure subsets of X (intuitively, those elements of Z that are not already in Σ are the ones preventing completeness from holding true);
let Σ0 be the σ-algebra generated by Σ and Z (i.e. the smallest σ-algebra that contains every element of Σ and of Z);
μ has an extension μ0 to Σ0 (which is unique if μ is σ-finite), called the outer measure of μ, given by the infimum
Then (X, Σ0, μ0) is a complete measure space, and is the completion of (X, Σ, μ).
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