Bierlein's measure extension theorem
Result in measure theory (2024-10-27 20:20:30)
by hozy@open.ibis.wiki
--- original
+++ modified
@@ -0,0 +1,7 @@
+In measure theory and probability theory, __Bierlein's measure extension theorem__ is a result about the extensibility of a probability measure. It states that a probability measure can be extended to the $\sigma$-algebra generated by the union of the underlying $\sigma$-algebra and an arbitrary family of disjoint subsets of the space. It has particular importance when dealing with infinite-dimensional spaces.
+
+## Statement
+
+The statement of Bierlein's measure extension theorem is
+
+> Let $(\Omega, \mathscr{F}, \mu)$ be a probability space and $(A_i)_{i\in I}$ a family of disjoint sets in $\Omega$ for some index set $I$. Then there is an extension of $\mu$ to $\sigma(\mathscr{F}\cup \{A_i\colon i\in I\})$.