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 $R$ be a ring of subsets (closed under union and relative complement) of a fixed set $X$ and let ${\mu }_0:R\to [0,\mathrm{\infty }]$ be a set function. ${\mu }_0$ is called a pre-measure if $${\mu }_0(\varnothing )=0$$ and, for every countable (or finite) sequence $A_1,A_2,\dots \in R$ of pairwise disjoint sets whose union lies in $R,$ $${\mu }_0\left(\bigcup _{n=1}^{\mathrm{\infty }}{A}_n\right)=\sum _{n=1}^{\mathrm{\infty }}{\mu }_0(A_n).$$ 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

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 $${\mu }^{*}(S)=\inf \{\sum _{i=1}^{\mathrm{\infty }}{\mu }_0(A_i)|A_i\in R,S\subseteq \bigcup _{i=1}^{\mathrm{\infty }}{A}_i\}$$ is an outer measure on $X$ and the measure $\mu$ induced by ${\mu }^{*}$ on the $\sigma$-algebra $\mathrm{\Sigma }$ of Carathéodory-measurable sets satisfies $\mu (A)={\mu }_0(A)$ for $A\in R$ (in particular, $\mathrm{\Sigma }$ includes $R$). The infimum of the empty set is taken to be $+\mathrm{\infty }.$

(Note that there is some variation in the terminology used in the literature. For example, Rogers (1998) uses "measure" and "pre-measure" where this article uses terms "outer measure" and "set function", respectively. Outer measures are not, in general, measures, since they may fail to be $\sigma$-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.