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)

Comments

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)