Pre-measure: Difference between revisions

Latest revision as of 19:10, 28 June 2022

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

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Additionally, a Template:Em is a [[pi-system\|Template:Pi-system]] where every complement $B ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ A Template:Em is a semiring where every complement $Ω ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ $A, B, A_{1}, A_{2}, \dots$ are arbitrary elements of $ℱ$ and it is assumed that $ℱ \neq \emptyset .$

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" where this article uses the term "outer measure". Outer measures are not, in general, measures, since they may fail to be $σ$ -additive.)

References

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Pre-measure: Difference between revisions

Latest revision as of 19:10, 28 June 2022

Contents

Definition

Carathéodory's extension theorem

See also

References

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Pre-measure: Difference between revisions

Latest revision as of 19:10, 28 June 2022

Definition

Carathéodory's extension theorem

See also

References

Navigation menu

Search