Results 1 to 2 of 2

Math Help - m(A {intersection} U {1 to infinity} Ei)=Sum {1 to infinity} m(A {intersection} Ei)

  1. #1
    Junior Member Dark Sun's Avatar
    Joined
    Apr 2009
    From
    San Francisco, California
    Posts
    32
    Thanks
    1

    m(A {intersection} U {1 to infinity} Ei)=Sum {1 to infinity} m(A {intersection} Ei)

    I have problem here which I am having difficulty proving:

    Assume that \langle E_1\rangle is a sequence of disjoint (Lebesgue) measurable sets, and A is any set. w.t.s.:

    m^*(A\cap \bigcup _{i=1}^{\infty } E_i)=\sum _{i=1}^{\infty }m^*(A\cap E_i)

    I have a proof of the finite case as i goes from 1 to n by using induction, however, it requires me to take the intersection of the last term in the sequence, which does not apply to the infinite case.

    Also, I believe that m^*(A\cap \bigcup _{i=1}^{\infty } E_i)\leq \sum _{i=1}^{\infty }m^*(A\cap E_i) is trivial since sub-additivity applies to the outer measure, and A\cap \bigcup _{i=1}^{\infty } E_i=\bigcup _{i=1}^{\infty }(A\cap E_i).

    So, it may only be neccessary to prove it going the other way.

    Thanks a million!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member Dark Sun's Avatar
    Joined
    Apr 2009
    From
    San Francisco, California
    Posts
    32
    Thanks
    1
    Hello, I apologize if my Analysis skills are lacking somewhat. But that's why I'm learning it, after all...

    Anyway, something came to me in a flash of insight, perhaps you could tell me if I am mistaken:

    It suffices to show that \exists N s.t. n>N implies that \forall \epsilon >0, \; |m^*(A\cap \bigcup _{i-1}^n\limits E_i)-\sum _{i=1}^n m^*(A\cap E_i)|<\epsilon .

    However, since m^*(A\cap \bigcup _{i-1}^n\limits E_i)=\sum _{i=1}^n m^*(A\cap E_i) by Lemma 9, we have that:

    |m^*(A\cap \bigcup _{i-1}^n\limits E_i)-\sum _{i=1}^n m^*(A\cap E_i)|=0<\epsilon
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: May 8th 2011, 05:13 AM
  2. Replies: 6
    Last Post: April 23rd 2011, 12:05 PM
  3. Replies: 3
    Last Post: February 1st 2011, 12:36 PM
  4. Limit at infinity equaling infinity?
    Posted in the Calculus Forum
    Replies: 6
    Last Post: October 1st 2010, 10:07 PM
  5. ilimts w/ result being infinity minus infinity
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 17th 2010, 05:18 AM

Search Tags


/mathhelpforum @mathhelpforum