Results 1 to 2 of 2

Thread: sigma-algebra generated by a r.v.

  1. April 30th 2012, 01:58 AM #1
    Juju
    Juju is offline
    Junior Member
    Joined
    Feb 2011
    Posts
    54
    Thanks
    1

    sigma-algebra generated by a r.v.

    Hallo,

    How can I show that the \sigma-algebra generated by a random variable X and the \sigma-algebra generated by X and f(X) ( f is an appropriate measurble function) are equal?

    I.e. I want to show

    \sigma(X)=\sigma(X,f(X)).

    Obviously, it holds that
    \sigma(f(X))\subseteq \sigma(X) \subseteq \sigma(X,f(X)).

    Can anybody help me?
    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+
  2. April 30th 2012, 02:02 AM #2
    girdav
    girdav is offline
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    674
    Thanks
    32

    Re: sigma-algebra generated by a r.v.

    Show that \sigma(X) contains all the elements of the form X^{-1}(B) and (f\circ X)^{-1}(B), where B is measurable.
    Follow Math Help Forum on Facebook and Google+
« Is a probability measure determined by the values it assigns to the closed intervals? | Infinite expectation with finite conditional expected value »

Similar Math Help Forum Discussions

  1. generated algebra
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: March 8th 2011, 12:28 PM
  2. Is a countable union/intersection of sigma-algebra also a sigma-algebra?
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 4th 2011, 08:39 AM
  3. finitely generated module & algebra
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 4th 2010, 03:39 AM
  4. 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
  5. Two linear algebra questions (finitely generated, isomorphism)
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: September 19th 2010, 03:30 PM

Search Tags


/mathhelpforum @mathhelpforum