, 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
of) metric spaces
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.
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
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.)
- Munroe, M. E. (1953). Introduction to measure and integration. Cambridge, Mass.: Addison-Wesley Publishing Company Inc..
- Rogers, C. A. (1998). Hausdorff measures. Third edition, Cambridge: Cambridge University Press. ISBN 0-521-62491-6. (See section 1.2.)