Pre-measure

In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.

Definition

Let [math]\displaystyle{ R }[/math] be a ring of subsets (closed under union and relative complement) of a fixed set [math]\displaystyle{ X }[/math] and let [math]\displaystyle{ \mu_0 : R \to [0, \infty] }[/math] be a set function. [math]\displaystyle{ \mu_0 }[/math] is called a pre-measure if [math]\displaystyle{ \mu_0(\varnothing) = 0 }[/math] and, for every countable (or finite) sequence [math]\displaystyle{ A_1, A_2, \ldots \in R }[/math] of pairwise disjoint sets whose union lies in [math]\displaystyle{ R, }[/math] [math]\displaystyle{ \mu_0 \left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu_0(A_n). }[/math] The second property is called [math]\displaystyle{ \sigma }[/math]-additivity.

Thus, what is missing for a pre-measure to be a measure is that it is not necessarily defined on a sigma-algebra (or a sigma-ring).

Carathéodory's extension theorem

It turns out that pre-measures give rise quite naturally to outer measures, which are defined for all subsets of the space [math]\displaystyle{ X. }[/math] More precisely, if [math]\displaystyle{ \mu_0 }[/math] is a pre-measure defined on a ring of subsets [math]\displaystyle{ R }[/math] of the space [math]\displaystyle{ X, }[/math] then the set function [math]\displaystyle{ \mu^* }[/math] defined by [math]\displaystyle{ \mu^* (S) = \inf \left\{\left. \sum_{i=1}^{\infty} \mu_0(A_i) \right| A_i \in R, S \subseteq \bigcup_{i=1}^{\infty} A_i\right\} }[/math] is an outer measure on [math]\displaystyle{ X }[/math] and the measure [math]\displaystyle{ \mu }[/math] induced by [math]\displaystyle{ \mu^* }[/math] on the [math]\displaystyle{ \sigma }[/math]-algebra [math]\displaystyle{ \Sigma }[/math] of Carathéodory-measurable sets satisfies [math]\displaystyle{ \mu(A) = \mu_0(A) }[/math] for [math]\displaystyle{ A \in R }[/math] (in particular, [math]\displaystyle{ \Sigma }[/math] includes [math]\displaystyle{ R }[/math]). The infimum of the empty set is taken to be [math]\displaystyle{ +\infty. }[/math]

(Note that there is some variation in the terminology used in the literature. For example, Rogers (1998) uses "measure" where this article uses the term "outer measure". Outer measures are not, in general, measures, since they may fail to be [math]\displaystyle{ \sigma }[/math]-additive.)

References

• Munroe, M. E. (1953). Introduction to measure and integration. Cambridge, Mass.: Addison-Wesley Publishing Company Inc.. pp. 310.  MR0053186
• Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. 195. ISBN 0-521-62491-6.  MR1692618 (See section 1.2.)
• Folland, G. B. (1999). Real Analysis. Pure and Applied Mathematics (Second ed.). New York: John Wiley & Sons, Inc. pp. 30–31. ISBN 0-471-31716-0. https://archive.org/details/realanalysismode00foll.