Bierlein's measure extension theorem
Fix math code (2024-10-27 20:22:45)
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-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.
+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 $\sigma(\mathscr{F}\cup \{A_i\colon i\in I\})$.
+> 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 $$\sigma(\mathscr{F}\cup \{A_i\colon i\in I\})$$.