Dynkin System - Definitions

Definitions

Let Ω be a nonempty set, and let be a collection of subsets of Ω (i.e., is a subset of the power set of Ω). Then is a Dynkin system if

  1. Ω ∈ ,
  2. if A, B ∈ and A ⊆ B, then B \ A ∈ ,
  3. if A1, A2, A3, ... is a sequence of subsets in and An ⊆ An+1 for all n ≥ 1, then .

Equivalently, is a Dynkin system if

  1. Ω ∈ ,
  2. if A ∈ D, then Ac ∈ ,
  3. if A1, A2, A3, ... is a sequence of subsets in such that Ai ∩ Aj = Ø for all i ≠ j, then .

An important fact is that a Dynkin system which is also a π-system (i.e., closed under finite intersection) is a σ-algebra. This can be verified by noting that condition 3 and closure under finite intersection implies closure under countable unions.

Given any collection of subsets of, there exists a unique Dynkin system denoted which is minimal with respect to containing . That is, if is any Dynkin system containing, then . is called the Dynkin system generated by . Note . For another example, let and ; then .

Read more about this topic:  Dynkin System

Famous quotes containing the word definitions:

    “What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
    —G.C. (Georg Christoph)

    “Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
    There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
    —Edmond De Goncourt (1822–1896)

    “The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
    —Pamela Patrick Novotny (20th century)