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. 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 of a fixed set X and let μ0: R → be a set function. μ0 is called a pre-measure if
and, for every countable sequence n∈N ⊆ R of pairwise disjoint sets whose union lies in R,
The second property is called σ-additivity.
Thus, what is missing for a pre-measure to be a measure is that it is not necessarily defined on a sigma-algebra.

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 μ0 is a pre-measure defined on a ring of subsets R of the space X, then the set function μ∗ defined by
is an outer measure on X and the measure μ induced by μ∗ on the σ-algebra Σ of Carathéodory-measurable sets satisfies for . The infimum of the empty set is taken to be.