Meaning of $\bigvee$ for collection of sets

Question

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.

Which book? It might not be a standard notation, but a notation the author is accustomed to using. — bjd2385, Oct 15 '15 at 14:14

GEdgar · Accepted Answer · 2015-10-15T14:31:01.707

2

$\bigvee_{i}\mathfrak{B}(X_i)$ 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.

edited Oct 15 '15 at 14:31

answered Oct 15 '15 at 14:21

GEdgar

111,679

And by generated we mean Borel, and then, in agreement with what you write below, the notation stands for the smallest $\sigma$-algebra containing $\bigcup_i \mathfrak{B}(X_i)$ ? – Three Diag Oct 15 '15 at 14:28
I see, Borel requires also to contain an associated topology. Thanks. – Three Diag Oct 15 '15 at 14:34
The author of the book refers to it also as the product sigma-algebra – Three Diag Oct 19 '15 at 15:55

Meaning of $\bigvee$ for collection of sets

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