Pre-measure
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a pre-measure is a
set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R ...
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 In set theory, the complement of a set , often denoted by A^c (or ), is the set of elements not in . When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set , the absolute complement ...
) of a fixed set X and let \mu_0 : R \to , \infty/math> be a
set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R ...
. \mu_0 is called a pre-measure if \mu_0(\varnothing) = 0 and, for every countable (or finite) sequence A_1, A_2, \ldots \in R of
pairwise disjoint In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
sets whose union lies in R, \mu_0 \left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu_0(A_n). The second property is called \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 measure In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer me ...
s, which are defined for all subsets of the space X. More precisely, if \mu_0 is a pre-measure defined on a ring of subsets R of the space X, then the set function \mu^* defined by \mu^* (S) = \inf \left\ is an outer measure on X and the measure \mu induced by \mu^* on the \sigma-algebra \Sigma of Carathéodory-measurable sets satisfies \mu(A) = \mu_0(A) for A \in R (in particular, \Sigma 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 \sigma-additive.)


See also

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References

* * (See section 1.2.) * {{Measure theory Measures (measure theory)