Bierlein's measure extension theorem
Bierlein's measure extension theorem is a result from measure theory and probability theory on extensions of probability measures. The theorem makes a statement about when one can extend a probability measure to a larger σ-algebra. It is of particular interest for infinite dimensional spaces.
The theorem is named after the German mathematician Dietrich Bierlein, who proved the statement for countable families in 1962.[1] The general case was shown by Albert Ascherl and Jürgen Lehn in 1977.[2]
A measure extension theorem of Bierlein
[edit]Let be a probability space and be a σ-algebra, then in general can not be extended to . For instance when is countably infinite, this is not always possible. Bierlein's extension theorem says, that it is always possible for disjoint families.
Statement of the theorem
[edit]Bierlein's measure extension theorem is
- Let be a probability space, an arbitrary index set and a family of disjoint sets from . Then there exists a extension of on .
Related results and generalizations
[edit]Bierlein gave a result which stated an implication for uniqueness of the extension.[1] Ascherl and Lehn gave a condition for equivalence.[2]
Zbigniew Lipecki proved in 1979 a variant of the statement for group-valued measures (i.e. for "topological hausdorff group"-valued measures).[3]
References
[edit]- ^ a b Bierlein, Dietrich (1962). "Über die Fortsetzung von Wahrscheinlichkeitsfeldern". Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. 1 (1): 28–46. doi:10.1007/BF00531770.
- ^ a b Ascherl, Albert; Lehn, Jürgen (1977). "Two principles for extending probability measures". Manuscripta Math. 21 (21): 43–50. doi:10.1007/BF01176900.
- ^ Lipecki, Zbigniew (1980). "A generalization of an extension theorem of Bierlein to group-valued measures". Bulletin Polish Acad. Sci. Math. 28: 441–445.