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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 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

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:

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



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

See also


  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.