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]

Let ${\displaystyle (X,{\mathcal {A}},\mu )}${\displaystyle (X,{\mathcal {A}},\mu )} be a probability space and ${\displaystyle {\mathcal {S}}\subset {\mathcal {P}}(X)}${\displaystyle {\mathcal {S}}\subset {\mathcal {P}}(X)} be a σ-algebra, then in general ${\displaystyle \mu }${\displaystyle \mu } can not be extended to ${\displaystyle \sigma ({\mathcal {A}}\cup {\mathcal {S}})}${\displaystyle \sigma ({\mathcal {A}}\cup {\mathcal {S}})}. For instance when ${\displaystyle {\mathcal {S}}}${\displaystyle {\mathcal {S}}} is countably infinite, this is not always possible. Bierlein's extension theorem says, that it is always possible for disjoint families.

Bierlein's measure extension theorem is

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

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]