Complete measure

related topics
{math, number, function}
{rate, high, increase}
{village, small, smallsup}

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

Contents

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 (RBλ). We now wish to construct 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 λ2measure zero. So, "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, Σ0μ0) 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 subsets of μ-measure zero 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);
  • there is a unique extension μ0 of μ to Σ0 given by the infimum

Then (X, Σ0μ0) 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

  • Borel measure as defined on the Borel σ-algebra generated by the open intervals of the real line is not complete, and so the above completion procedure must be used to define the complete Lebesgue measure.
  • 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.

Full article ▸

related documents
LALR parser
Contraction mapping
Blum Blum Shub
Best-first search
Normal morphism
Nearest neighbour algorithm
Lyapunov fractal
Axiom of union
Sophie Germain prime
Baire category theorem
Mutual recursion
Composite number
Group homomorphism
Partition of unity
Characteristic subgroup
Linear congruence theorem
Up to
Hamiltonian path problem
Multiple inheritance
Permutation group
Complete category
Category of sets
Tree structure
Data set
Zhu Shijie
AVL tree
Elias delta coding
Sed
Brun's constant
Identifier