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 be a ring of subsets of a fixed set and let be a set function. is called a pre-measure ifand, for every countable (or finite) sequence of pairwise disjoint sets whose union lies in
The second property is called -additivity.
Thus, what is missing for a pre-measure to be a measure is that it is not necessarily defined on a sigma-algebra.
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 More precisely, if is a pre-measure defined on a ring of subsets of the space then the set function defined byis an outer measure on and the measure induced by on the -algebra of Carathéodory-measurable sets satisfies for . The infimum of the empty set is taken to be