Pre-measure

Template:Short description 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 $R$ be a ring of subsets (closed under union and relative complement) of a fixed set $X$ and let $μ_{0} : R \to [0, \infty]$ be a set function. $μ_{0}$ is called a pre-measure if $μ_{0} (\emptyset) = 0$ and, for every countable (or finite) sequence $A_{1}, A_{2}, \dots \in R$ of pairwise disjoint sets whose union lies in $R,$ $μ_{0} (⋃_{n = 1}^{\infty} A_{n}) = \sum_{n = 1}^{\infty} μ_{0} (A_{n}) .$ 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 (or a sigma-ring).

Carathéodory's extension theorem

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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 $μ_{0}$ is a pre-measure defined on a ring of subsets $R$ of the space $X,$ then the set function $μ^{*}$ defined by $μ^{*} (S) = inf {\sum_{i = 1}^{\infty} μ_{0} (A_{i}) | A_{i} \in R, S \subseteq ⋃_{i = 1}^{\infty} A_{i}}$ is an outer measure on $X$ and the measure $μ$ induced by $μ^{*}$ on the $σ$ -algebra $Σ$ of Carathéodory-measurable sets satisfies $μ (A) = μ_{0} (A)$ for $A \in R$ (in particular, $Σ$ includes $R$ ). The infimum of the empty set is taken to be $+ \infty .$

(Note that there is some variation in the terminology used in the literature. For example, Rogers (1998) uses "measure" and "pre-measure" where this article uses terms "outer measure" and "set function", respectively. Outer measures are not, in general, measures, since they may fail to be $σ$ -additive.)

References

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Template:Measure theory

Pre-measure

Contents

Definition

Carathéodory's extension theorem

See also

References

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Pre-measure

Definition

Carathéodory's extension theorem

See also

References

Navigation menu

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