# Pre-measure

A finitely-additive measure with real or complex values on some space $\Omega$ having the property that it is defined on an algebra $\mathfrak A$ of subsets of $\Omega$ of the form $\mathfrak A = \cup _ {\alpha \in A } \mathfrak B _ \alpha$, where $\mathfrak B _ \alpha$ is a family of $\sigma$-algebras of $\Omega$, labelled by the elements of some partially ordered set $A$, such that $\mathfrak B _ {\alpha _ {1} } \subset \mathfrak B _ {\alpha _ {2} }$ if $\alpha _ {1} < \alpha _ {2}$, while the restriction of the measure to any $\sigma$-algebra $\mathfrak B _ \alpha$ is countably additive. E.g., if $\Omega$ is a Hausdorff space, $A$ is the family of all compacta, ordered by inclusion, $\mathfrak B _ \alpha$, $\alpha \in A$, is the $\sigma$-algebra of all Borel subsets of the compactum $\alpha$ and $C _ {0} ( \Omega )$ is the space of all continuous functions on $\Omega$ with compact support, then every linear functional on $C _ {0} ( \Omega )$ that is continuous in the topology of uniform convergence in $C _ {0} ( \Omega )$ generates a pre-measure on the algebra $\mathfrak A = \cup _ {\alpha \in A } \mathfrak B _ \alpha$.

Let $\Omega$ be a locally convex linear space, let $A$ be the set of finite-dimensional subspaces of the dual space $\Omega ^ \prime$, ordered by inclusion, and let $\mathfrak B _ \alpha$, $\alpha \in A$, be the least $\sigma$-algebra relative to which all linear functionals $\phi \in \alpha$ are measurable. The sets of the algebra $\mathfrak A = \cup _ {\alpha \in A } \mathfrak B$ are called cylindrical sets, and any pre-measure on $\mathfrak A$ is called a cylindrical measure (or quasi-measure). A positive-definite functional on $\Omega ^ \prime$ that is continuous on any finite-dimensional subspace $\alpha \subset \Omega$ is the characteristic function (Fourier transform) of a finite non-negative pre-measure on $\Omega$.

#### References

 [1] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)

The term "pre-measure" is also used in the following, related but somewhat different, sense. Let ${\mathcal R}$ be a ring of sets on some space $\Omega$, and $\mu$ a numerical function defined on ${\mathcal R}$. Then $\mu$ is a pre-measure if

i) $\mu ( \emptyset ) = 0$, $\mu ( A) \geq 0$ for all $A \in {\mathcal R}$;

ii) $\mu ( \cup _ {n= 1} ^ \infty A _ {n} ) = \sum _ {n= 1} ^ \infty \mu ( A _ {n} )$ for every countable sequence of pairwise disjoint subsets $A _ {n} \in {\mathcal R}$ such that $\cup A _ {n} \in {\mathcal R}$.

If ii) only holds for finite disjoint sequences, $\mu$ is called a content. Not every content is a pre-measure.

#### References

 [a1] H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) pp. 13ff (Translated from German)
How to Cite This Entry:
Pre-measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-measure&oldid=52336
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article