Results 1 to 4 of 4

Math Help - convergence in measure

  1. #1
    Member
    Joined
    Dec 2008
    Posts
    154

    convergence in measure

    Let (X,F, \mu) be a measure space with \mu (X) < \infty.
    Prove that f_n \rightarrow f in measure \Longleftrightarrow \int_X \frac{|f_n-f|}{1+|f_n-f|} d \mu \rightarrow 0.
    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,

    See here : http://www.mathhelpforum.com/math-he...-1-yn-0-a.html

    (the integral is like the expectation, and measure = probability)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2008
    Posts
    154
    i understand the first half. but i have a question on the second half.
    by letting Y_n=|f_n-f|, Z_n = \frac{Y_n}{1+Y_n}and mimicking the steps. i have

    \int Z_n d\mu = \int Z_n 1_{\{Z_n \leq \epsilon \} } d\mu + \int Z_n 1_{ \{Z_n > \epsilon \} } d\mu

    \int Z_n 1_{\{Z_n \leq \epsilon \} } d\mu \leq \int \epsilon 1_{\{Z_n \leq \epsilon \} } d\mu = \epsilon \int 1_{\{Z_n \leq \epsilon \} } d\mu

    but how do i show that \epsilon \int 1_{\{Z_n \leq \epsilon \} } d\mu \leq \epsilon without knowing that \mu is a probability measure? is there anyway i can estimate integral \epsilon \int 1_{\{Z_n \epsilon \} }? please help me.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    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
    It's not that difficult, think about an hypothesis you've not been using

    Spoiler:
    \{Z_n<\epsilon\}\subset X \Rightarrow \mu(Z_n<\epsilon)=\int_X \bold{1}_{\{Z_n<\epsilon\}} ~d\mu \leq \mu(X) <\infty

    So you can say that it's <\epsilon' where \epsilon'=\epsilon/\mu(X)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Convergence in measure
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 10th 2010, 07:19 PM
  2. Convergence in measure
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 26th 2010, 07:14 AM
  3. Convergence in measure
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 25th 2010, 08:57 AM
  4. convergence in measure
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: January 2nd 2010, 12:44 AM
  5. dominated convergence theorem for convergence in measure
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: December 5th 2009, 04:06 AM

Search Tags


/mathhelpforum @mathhelpforum