Pre-measure

In mathematics, a pre-measure is a function that is, in some sense, a precursor to a bona fide measure on a given space. Pre-measures are particularly useful in fractal geometry and dimension theory, where they can be used to define measures such as Hausdorff measure and packing measure on (subsets of) metric spaces.

Definition

Let ${\displaystyle R}$ be a ring of sets and $\mu _{0}:R\rightarrow [0,\infty ]$ a set function. ${\displaystyle \mu _{0}}$ is called a pre-measure if

$\mu _{0}(\emptyset )=0$

and for a sequence $\{A_{n}\}_{n=1}^{\infty }\subseteq R$ of pairwise disjoint sets whose union lies in ${\displaystyle R}$

$\mu _{0}\left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu _{0}(A_{n})$.

The last property is called $\sigma$-additivity

Extension theorem

It turns out that pre-measures can be extended quite naturally to outer measures, which are defined for all subsets of the space ${\displaystyle X}$. More precisely, if ${\displaystyle \mu _{0}}$ is a pre-measure defined on a ring of subsets of the space ${\displaystyle X}$, then the set function ${\displaystyle \mu ^{*}}$ defined by

$\mu ^{*}(S)=\inf \left\{\left.\sum _{i=1}^{\infty }\mu _{0}(S_{i})\right|S_{i}\in X,S\subseteq \bigcup _{i=1}^{\infty }S_{i}\right\}$

is an outer measure on ${\displaystyle X}$.

(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 σ-additive.)

See also

• Hahn-Kolmogorov theorem

References

• Munroe, M. E. (1953). Introduction to measure and integration. Cambridge, Mass.: Addison-Wesley Publishing Company Inc. p. 310. MR0053186
• Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. p. 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.