The need toconsiderquestions of completeness can be illustrated by considering the problem of productspaces. 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 λ2 on the planeR2 as a product measure. Naively, 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 × 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 measure space, there is an extension of this measure space that is complete. The smallest such extension 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 ;
Then is a complete measure space, and is the completion of. 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
Borel measure as defined on the Borel σ-algebra generated by the openintervals of the real line is not complete, and so the above completion procedure must be used to define the complete Lebesgue measure. This is illustrated by the fact that the set of all Borel sets over the reals has the same cardinality as the reals. While the Cantor set is a Borel set, has measure zero, and its power set has cardinality strictly greater than that of the reals. Thus there is a subset of the Cantor set that is not contained in the Borel sets. Hence, the Borel measure is not complete.
n-dimensional Lebesgue measure is the completion of the n-fold product of the one-dimensional Lebesgue space with itself. It is also the completion of the Borel measure, as in the one-dimensional case.
