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Filtered probability space

  1. Oct 25, 2013 #1
    atkpqw.jpg
     
  2. jcsd
  3. Oct 25, 2013 #2
    For a collection [itex]\{\mathcal F_s\}_{s\in S}[/itex] of sub-[itex]\sigma[/itex]-algebras of [itex]\mathcal F[/itex], the set [itex]\bigvee_{s\in S} \mathcal F_s[/itex] is defined to be the smallest sub-[itex]\sigma[/itex]-algebra [itex]\mathcal G\subseteq \mathcal F[/itex] such that [itex]\mathcal F_s\subseteq\mathcal G[/itex] for every [itex]s\in S[/itex].
     
  4. Oct 26, 2013 #3
    at http://en.wikipedia.org/wiki/Filtration_(mathematics)#Measure_theory

    the encircle part says:

    rmnzph.jpg

    Similarly in the following picture the encircle part

    2ag5bnl.jpg

    does the symbole " 10fuquf.jpg " 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)?
     
  5. Oct 26, 2013 #4
    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.
     
  6. Oct 26, 2013 #5
    Thanks
     
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