Complete measure

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a complete measure (or, more precisely, a complete measure space) is a measure space

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 in which every subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

 of every null set

Null set

In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...

 is measurable (having measure zero). More formally, (X, Σ, μ) is complete if and only if

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

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

 on the real line

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

: denote this measure space by (R, B, λ). We now wish to construct two-dimensional Lebesgue measure λ² on the plane R² as a product measure

Product measure

In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space...

. Naïvely, we would take the σ-algebra on R² to be B ⊗ B, the smallest σ-algebra containing all measurable "rectangles" A₁ × A₂ for A_i ∈ 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

Non-measurable set

In mathematics, a non-measurable set is a set whose structure is so complicated that it cannot be assigned any meaningful measure. Such sets are constructed to shed light on the notions of length, area and volume in formal set theory....

 of the real line, such as the Vitali set

Vitali set

In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...

. 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, 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 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
  Infimum
  In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
  
  

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

Examples

• Borel measure as defined on the Borel σ-algebra generated by the open
  Open set
  The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
  
   intervals
  Interval (mathematics)
  In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
  
  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.