Results 1 to 3 of 3

Math Help - Sigma - fields

  1. #1
    Newbie
    Joined
    Mar 2010
    Posts
    2

    Sigma - fields

    Hi all,

    need a bit of help on this question.

    Let

    omega = {1, . . . , 6} and A = {{1, 3, 5}, {1, 2, 3}}.
    a) Describe F = sigma(A), the sigma-field generated by A.
    Hint: For a finite set omega
    , the number of elements of a sigma-field on omega
    is always a power of 2.

    I have grasped the concept that F denotes a collection of subsets on omega, and also A is a collection of events that generates a sigma-field. However full appreciation of this concept hasn't sunk in yet.

    Is there a smart soul out there who could please enlighten me?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    The hint is here to let you know if you've forgotten any element.
    We know that the empty set and omega belong to the sigma-field. That makes 2.

    {1,3,5} and {1,2,3} obviously belong to the sigma-field, since they generate it.

    We know a sigma-field is closed under union. So {1,3,5} U {1,2,3}={1,2,3,5} belong to the sigma-field.

    Then we know it's closed under the complement : find the complement of {1,3,5}, {1,2,3} and {1,2,3,5} in Omega : {2,4,6}, {4,5,6}, {4,6}. That makes 3 more.

    But now you can also do {1,3,5} U {4,5,6}={1,3,4,5,6} and its complement {2}
    {1,2,3} U {2,4,6}={1,2,3,4,6} and its complement {5}
    {2,4,6} U {4,5,6}={2,4,5,6} and its complement {1,3}

    And the remaining union : {2} U {5}={2,5} and its complement {1,3,4,6}

    That makes 16=2^4


    You have to try finding all the possibilities...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2010
    Posts
    2
    Thanks for such a clear answer. Wow.......Deeply appreciated!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lebesgue measure on sigma-fields of Borel sets
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 27th 2011, 07:56 PM
  2. Measurability of of Random Variables w.r.t. Sigma Fields
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: September 21st 2011, 06:07 PM
  3. Show intersection of sigma-algebras is again a sigma-algebra
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 20th 2010, 07:21 AM
  4. how to determine sigma and sgn(sigma) for matrices
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 16th 2010, 11:12 AM
  5. Extension fields / splitting fields proof...
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 19th 2007, 07:29 AM

Search Tags


/mathhelpforum @mathhelpforum