Pre-measure explained

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

R be a ring of subsets (closed under union and relative complement) of a fixed set X and let \mu _{0} : R \to [ 0 , inf ty] be a set function. \mu _{0} is called a pre-measure if $$\backslash mu\_0(\backslash varnothing)\; =\; 0$$and, for every countable (or finite) sequence A _{1,} A _{2,} \ldots \in R of pairwise disjoint sets whose union lies in R, $$\backslash mu\_0\; \backslash left(\backslash bigcup\_^\backslash infty\; A\_n\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash mu\_0(A\_n).$$The second property is called \sigma -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

See main article: 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

X. More precisely, if \mu _{0} is a pre-measure defined on a ring of subsets R of the space X, then the set function \mu ^{*} defined by$$\backslash mu^*\; (S)\; =\; \backslash inf\; \backslash left\backslash$$is an outer measure on X and the measure \mu induced by \mu ^{*} on the \sigma -algebra \Sigma of Carathéodory-measurable sets satisfies \mu(A) = \mu _{0(A)} for A \in R (in particular, \Sigma includes R ). The infimum of the empty set is taken to be +inf ty.

(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

\sigma -additive.)

References

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