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The axioms of sigma-agebras

  1. Oct 14, 2006 #1

    quasar987

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    Wiki says that a sigma algebra (or sigma field) is a subset [itex]\Sigma[/itex] of the powerset of some set X satisfying the following axioms

    1) [tex]E\in \Sigma \Rightarrow E^c \in \Sigma[/tex]

    2) [tex]E_i \in \Sigma \ \ \forall i \in I \Rightarrow \bigcup_{i\in I}E_i \in \Sigma[/tex]

    (where the index set I is countable)

    Am I missing something or is axiom 2 equivalent to the much less complicated "2') [itex]X\in \Sigma[/itex]"? Cause for any element of [itex]\Sigma[/itex], since its complement is in [itex]\Sigma[/itex] also, the union of both is X itself. So 2) is satified as soon as 2') is. Conversely, 2) implies that X is in [itex]\Sigma[/itex] simply by taking an element of [itex]\Sigma[/itex] and its complement in the union.
     
    Last edited: Oct 14, 2006
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  3. Oct 14, 2006 #2

    Hurkyl

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    (2) applies to any sequence of elements of sigma. That includes sequences that don't contain a pair of complementary sets.
     
  4. Oct 14, 2006 #3

    quasar987

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    oh, right!
     
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