What is a Filtered Probability Space?

[woundedtiger4]

 

[economicsnerd]

For a collection $\{\mathcal F_s\}_{s\in S}$ of sub-$\sigma$-algebras of $\mathcal F$, the set $\bigvee_{s\in S} \mathcal F_s$ is defined to be the smallest sub-$\sigma$-algebra $\mathcal G\subseteq \mathcal F$ such that $\mathcal F_s\subseteq\mathcal G$ for every $s\in S$.

 

[woundedtiger4]

For a collection $\{\mathcal F_s\}_{s\in S}$ of sub-$\sigma$-algebras of $\mathcal F$, the set $\bigvee_{s\in S} \mathcal F_s$ is defined to be the smallest sub-$\sigma$-algebra $\mathcal G\subseteq \mathcal F$ such that $\mathcal F_s\subseteq\mathcal G$ for every $s\in S$.

at http://en.wikipedia.org/wiki/Filtration_(mathematics)#Measure_theory

the encircle part says:

Similarly in the following picture the encircle part

does the symbole "

" represent Union? I understood the notation from your reply but what is the name of the symbol? is it logical disjunction (though logical disjunction doesn't make sense here) or is it universal quantifier (though logical quantifier is turned A) or is it just a capital V (the 22nd alphabet)?

 

[R136a1]

The symbol is called a "join". It is a symbol from lattice theory. Here, just means the smallest sigma-algebra that contains the union.

The dual symbol is $\bigwedge$ and is called a meet.

 

[woundedtiger4]

Thanks