1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Measure space, null set

  1. May 24, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex](X,\mathcal{A},\mu)[/tex] be a fixed measure space.

    Let [tex]A_k \in \mathcal{A}[/tex] such that [tex]\displaystyle
    \sum^\infty_{k=1} \mu(A_k) < \infty[/tex]. Prove that
    [tex]
    \begin{align*}
    \{ x \in X | x \in A_k \text{ for infinitely many k} \}
    \end{align*}
    [/tex]
    is a null set.

    2. Relevant equations



    3. The attempt at a solution

    Let [tex]S = \{ x \in X | x \in A_k \text{ for infinitely many k} \}[/tex].
    Suppose [tex]\mu (S) > 0[/tex]. Then [tex]\displaystyle \mu(\bigcap A_k) > 0, A_k
    \ni x, x \in S[/tex]. Then [tex]\mu (A_k) > 0, A_k \ni x, x \in S[/tex] and hence
    [tex]\displaystyle \sum^\infty_{k=1} \mu(A_k) = \infty[/tex], which
    contradicts to assumption.

    Is this correct?
     
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted