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.{{cite journal |first=Dietrich |last=Bierlein |title=Über die Fortsetzung von Wahrscheinlichkeitsfeldern |journal=Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete |number=1 |pages=28–46 |date=1962|volume=1 |doi=10.1007/BF00531770 }} The general case was shown by Albert Ascherl and Jürgen Lehn in 1977.{{cite journal |first1=Albert |last1=Ascherl |first2=Jürgen |last2=Lehn |title=Two principles for extending probability measures |journal=Manuscripta Math. |number=21 |pages=43–50 |date=1977|volume=21 |doi=10.1007/BF01176900 }}

A measure extension theorem of Bierlein

Let $(X,\mathcal{A},\mu)$ be a probability space and $\mathcal{S}\subset \mathcal{P}(X)$ be a σ-algebra, then in general $\mu$ can not be extended to $\sigma(\mathcal{A}\cup \mathcal{S})$ . For instance when $\mathcal{S}$ 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 =

Bierlein's measure extension theorem is

:Let $(X,\mathcal{A},\mu)$ be a probability space, $I$ an arbitrary index set and $(A_i)_{i\in I}$ a family of disjoint sets from $X$ . Then there exists a extension $\nu$ of $\mu$ on $\sigma(\mathcal{A}\cup\{A_i\colon i\in I\} )$ .

= Related results and generalizations =

Bierlein gave a result which stated an implication for uniqueness of the extension. Ascherl and Lehn gave a condition for equivalence.

Zbigniew Lipecki proved in 1979 a variant of the statement for group-valued measures (i.e. for "topological hausdorff group"-valued measures).{{cite journal|first=Zbigniew |last=Lipecki |title=A generalization of an extension theorem of Bierlein to group-valued measures |journal=Bulletin Polish Acad. Sci. Math. |volume=28 |pages=441–445 |date=1980}}

References

Category:Theorems in measure theory

Category:Theorems in probability theory