In a book* I found the following:
$\mathcal{F} = \bigvee_i \mathfrak{B}(X_i)$
where $\mathcal{F}$ denotes a $\sigma $-algebra on a markov chain and $\mathfrak{B}(X_i)$ is the Borel sigma algebra generated by the set on which the random variables that constitute the chain take values.
What is the exact meaning of the $\bigvee_i$ notation? How does it relates or is different from the concept of union? And ultimately, why is it used in the context of measure theory and markov chains?
A similar question is here, but it refers to propositions rather than collection of sets.
I can see that a set is a proposition, and I now that $\vee$ is used in logic to denote inclusive or, but then why not just union?
- EDIT: Book is Markov Chains and Stochastic Stability by Meyn and Tweedie. Unfortunately, the symbol is not mentioned anywhere beforehand nor it is mentioned in the list of symbols.
$\bigvee_{i}\mathfrak{B}(X_i)$ means least sigma-algebra containing the union $\bigcup_{i}\mathfrak{B}(X_i)$.
Generally, $\bigvee$ and $\bigwedge$ are used in partially ordered sets for the least upper bound and greatest lower bound. In this case, the collection of $\sigma$-algebras on a fixed set is your partially ordered set.