Bierlein's measure extension theorem explained

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

Let

(X, l{A},\mu) be a probability space and l{S}\subset l{P}(X) be a σ-algebra, then in general \mu can not be extended to \sigma( l{A}\cup l{S}) . For instance when l{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, l{A},\mu) be a probability space, I an arbitrary index set and (A _{i)} _{i\in} a family of disjoint sets from X . Then there exists a extension \nu of \mu on \sigma( l{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).^[3]

References

1. Dietrich . Bierlein . Über die Fortsetzung von Wahrscheinlichkeitsfeldern . Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete . 1 . 28–46 . 1962. 1 . 10.1007/BF00531770 .
2. Albert . Ascherl . Jürgen . Lehn . Two principles for extending probability measures . Manuscripta Math. . 21 . 43–50 . 1977. 21 . 10.1007/BF01176900 .
3. Zbigniew . Lipecki . A generalization of an extension theorem of Bierlein to group-valued measures . Bulletin Polish Acad. Sci. Math. . 28 . 441–445 . 1980.