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, \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, \ldots \in R of pairwise disjoint sets whose union lies in R,
\mu_0 \left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\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
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 \left\{\left. \sum_{i=1}^{\infty} \mu_0(A_i) \right| A_i \in R, S \subseteq \bigcup_{i=1}^{\infty} A_i\right\}
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 +\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. p. 310.
- Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library. Third ed. Cambridge: Cambridge University Press. p. 195. ISBN 0-521-62491-6. (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.