Pre-measure

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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.

[edit] Definition

Let X be any set. A function p defined on a class C of subsets of X is said to be a pre-measure if

  • the collection C contains the empty set, ∅;
  • the function p assumes only non-negative (but possibly infinite) values: for all S ∈ C, 0 ≤ p(S) ≤ +∞;
  • the pre-measure of the empty set is zero: p(∅) = 0.

[edit] 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 X. More precisely, if p is a pre-measure defined on a class C of subsets of X, then the set function μ defined by

\mu (S) = \inf \left\{ \left. \sum_{i = 1}^{\infty} p(S_{i}) \right| S_{i} \in C, S \subseteq \bigcup_{i = 1}^{\infty} S_{i} \right\}

is an outer measure on 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.)

[edit] References

  • Munroe, M. E. (1953). Introduction to measure and integration. Cambridge, Mass.: Addison-Wesley Publishing Company Inc., pp. x+310.  MR0053186
  • Rogers, C. A. (1998). Hausdorff measures, Third edition, Cambridge Mathematical Library, Cambridge: Cambridge University Press, pp. xxx+195. ISBN 0-521-62491-6.  MR1692618 (See section 1.2.)