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 {\displaystyle R} be a ring of subsets (closed under union and relative complement) of a fixed set X {\displaystyle X} and let μ 0 : R → [ 0 , ∞ ] {\displaystyle \mu _{0}:R\to [0,\infty ]} be a set function. μ 0 {\displaystyle \mu _{0}} is called a pre-measure if
μ 0 ( ∅ ) = 0 {\displaystyle \mu _{0}(\varnothing )=0}
and, for every countable (or finite) sequence A 1 , A 2 , … ∈ R {\displaystyle A_{1},A_{2},\ldots \in R} of pairwise disjoint sets whose union lies in R , {\displaystyle R,}
μ 0 ( ⋃ n = 1 ∞ A n ) = ∑ n = 1 ∞ μ 0 ( A n ) . {\displaystyle \mu _{0}\left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu _{0}(A_{n}).}
The second property is called σ {\displaystyle \sigma } -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
It turns out that pre-measures give rise quite naturally to outer measures, which are defined for all subsets of the space X . {\displaystyle X.} More precisely, if μ 0 {\displaystyle \mu _{0}} is a pre-measure defined on a ring of subsets R {\displaystyle R} of the space X , {\displaystyle X,} then the set function μ ∗ {\displaystyle \mu ^{*}} defined by
μ ∗ ( S ) = inf { ∑ i = 1 ∞ μ 0 ( A i ) | A i ∈ R , S ⊆ ⋃ i = 1 ∞ A i } {\displaystyle \mu ^{*}(S)=\inf \left\{\left.\sum _{i=1}^{\infty }\mu _{0}(A_{i})\right|A_{i}\in R,S\subseteq \bigcup _{i=1}^{\infty }A_{i}\right\}}
is an outer measure on X {\displaystyle X} and the measure μ {\displaystyle \mu } induced by μ ∗ {\displaystyle \mu ^{*}} on the σ {\displaystyle \sigma } -algebra Σ {\displaystyle \Sigma } of Carathéodory-measurable sets satisfies μ ( A ) = μ 0 ( A ) {\displaystyle \mu (A)=\mu _{0}(A)} for A ∈ R {\displaystyle A\in R} (in particular, Σ {\displaystyle \Sigma } includes R {\displaystyle R} ). The infimum of the empty set is taken to be + ∞ . {\displaystyle +\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 {\displaystyle \sigma } -additive.)
See also Hahn-Kolmogorov theorem – Theorem extending pre-measures to measuresPages displaying short descriptions of redirect targets
References
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