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Real Variables: Measurability of {x: x∈An i.o.}

  1. Sep 25, 2011 #1
    1. The problem statement, all variables and given/known data

    Let An, n = 1,2,..., be a sequence of measurable sets. Let E = {x: x∈An i.o.}.

    (a) Prove that E is a measurable set.

    (b) Prove that m(E) = 0 if ∑m(An) < ∞


    2. Relevant equations

    A point x is said to be in An infinitely often (i.o.) if there is an infinite sequence of integers n1<n2<... such that x∈Ank for every k.


    3. The attempt at a solution

    I'm really not sure where to start with part (a). For part (b), if ∑m(An) < ∞
    then E is countable, therefore m(E) = 0...I can't really explain why E is countable, though, it's just an instinct.

    Any hints would be greatly appreciated : )
     
  2. jcsd
  3. Sep 25, 2011 #2

    CompuChip

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    Homework Helper

    My first instinct would be to try and explicitly define the measure on E.
    Each [itex]A_n[/itex] comes with its own measure [itex]\mu_n[/itex] so you could try something like
    [tex]\mu_E(x) := \sum_{n \mid x \in A_n} \mu_n(x)[/tex]
    and check that it is a measure.
     
  4. Sep 25, 2011 #3
    I think I figured it out. Thank you!
     
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