Bierlein's measure extension theorem

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

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]