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] 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
as a product measure.
the smallest 𝜎-algebra containing all measurable "rectangles"
While this approach does define a measure space, it has a flaw.
Since every singleton set has one-dimensional Lebesgue measure zero,
is a non-measurable subset of the real line, such as the Vitali set.
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.
[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: 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 Maharam's theorem states that every complete measure space is decomposable into measures on continua, and a finite or countable counting measure.