Complete 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 edit

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 edit

Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ0μ0) 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, 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 edit

  • 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. 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.

Properties edit

Maharam's theorem states that every complete measure space is decomposable into measures on continua, and a finite or countable counting measure.

See also edit

References edit

  • Terekhin, A.P. (2001) [1994], "Complete measure", Encyclopedia of Mathematics, EMS Press
  1. ^ Halmos, Paul R. (1950). Measure Theory. Graduate Texts in Mathematics. Vol. 18. New York, NY: Springer New York. p. 31. doi:10.1007/978-1-4684-9440-2. ISBN 978-1-4684-9442-6.
  2. ^ de Barra, G. (2003). Measure theory and integration. Woodhead Publishing Limited. p. 94. doi:10.1533/9780857099525. ISBN 978-1-904275-04-6.
  3. ^ Rudin, Walter (2013). Real and complex analysis. McGraw-Hill international editions Mathematics series (3. ed., internat. ed., [Nachdr.] ed.). New York, NY: McGraw-Hill. pp. 27–28. ISBN 978-0-07-054234-1.