Measure space where every subset of a set with null measure is measurable (and has null measure)
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[1][2]
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 on the plane as a product measure. Naively, we would take the 𝜎-algebra on 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 any 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.
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.[3] 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).
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
Properties
Maharam's theorem states that every complete measure space is decomposable into measures on continua, and a finite or countable counting measure.