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Real analysis: Problem similar to uniformly integrable

  1. Nov 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Assume [itex]\mu[/itex](X) >0 and that f is a measurable function that maps X into ℝ and satisfies f(x) >0 for all x[itex]\in[/itex]X.

    Let [itex]\alpha[/itex] be any fixed real number satisfying 0<[itex]\alpha[/itex]<[itex]\mu[/itex](X) <infinity Prove that

    inf { [itex]\int[/itex][itex]_{E}[/itex]f d[itex]\mu[/itex] : E[itex]\in[/itex]M, [itex]\mu[/itex](E) ≥[itex]\alpha[/itex]} >0.

    (Hint. First prove for any [itex]\delta[/itex] satisfying 0<[itex]\delta[/itex]<[itex]\alpha[/itex], prov there exists n [itex]\in[/itex] N such that B[itex]_{n}[/itex] = {x:f(x)[itex]\geq[/itex]1/n} satisfies [itex]\mu[/itex](B[itex]_{n}[/itex]) [itex]\geq[/itex]
    [itex]\mu[/itex](X) - [itex]\delta[/itex]. Then prove that if [itex]\mu[/itex](E) ≥[itex]\alpha[/itex] then [itex]\mu[/itex](E[itex]\cap[/itex]B[itex]_{n}[/itex]) ≥[itex]\alpha[/itex]-[itex]\delta[/itex]

    2. Relevant equations

    Not sure what is relevant.

    3. The attempt at a solution

    So, the hint suggests the first thing I do is show that given 0<[itex]\delta[/itex]<[itex]\alpha[/itex] i can find an n so that [itex]\mu[/itex](B[itex]_{n}[/itex]) [itex]\geq[/itex] [itex]\mu[/itex](X) - [itex]\delta[/itex]. So since f(x) >0 then [itex]\mu[/itex](X) = [itex]\mu[/itex]({x:f(x)>0}) = [itex]\mu[/itex]({x:f(x)≥1/n} [itex]\cup[/itex]{x:f(x)<1/n}). These are disjoint so we have [itex]\mu[/itex]({x:f(x)≥1/n}) + [itex]\mu[/itex]({x:f(x)<1/n}). If i show the delta relation with [itex]\mu[/itex]({x:f(x)<1/n}) then i get [itex]\mu[/itex](B[itex]_{n}[/itex]) [itex]\geq[/itex] [itex]\mu[/itex](X) - [itex]\delta[/itex]. It seems really trivial though and im not sure what guarantees this.

    The next part asks you to show that [itex]\mu[/itex](E[itex]\cap[/itex]B[itex]_{n}[/itex]) ≥[itex]\alpha[/itex]-[itex]\delta[/itex]. Ok so [itex]\mu[/itex](E[itex]\cap[/itex]B[itex]_{n}[/itex]) = [itex]\mu[/itex](E) +[itex]\mu[/itex](B[itex]_{n}[/itex]) - [itex]\mu[/itex](E[itex]\cup[/itex]B[itex]_{n}[/itex])

    The following hold: [itex]\mu[/itex](E[itex]\cup[/itex]B[itex]_{n}[/itex]) ≤[itex]\mu[/itex](X) (both are subsets)
    [itex]\mu[/itex](B[itex]_{n}[/itex]) ≥ [itex]\mu[/itex](X) - [itex]\delta[/itex] (previous part)
    [itex]\mu[/itex](E)≥[itex]\alpha[/itex] (hypothesis)

    So [itex]\mu[/itex](E[itex]\cap[/itex]B[itex]_{n}[/itex]) = [itex]\mu[/itex](E) +[itex]\mu[/itex](B[itex]_{n}[/itex]) - [itex]\mu[/itex](E[itex]\cup[/itex]B[itex]_{n}[/itex]) ≥ [itex]\mu[/itex](X) - [itex]\delta[/itex] + [itex]\mu[/itex](E) - [itex]\mu[/itex](X) =[itex]\mu[/itex](E)- [itex]\delta[/itex]≥[itex]\alpha[/itex]-[itex]\delta[/itex]


    So I've done most of what the hint wants... the problem is I don't know how this helps me.
     
  2. jcsd
  3. Nov 23, 2012 #2
    The title is misleading, I thought it related to uniformly intagrable but I don't think it does.
     
  4. Nov 24, 2012 #3
    I figured out why there is an n for every δ. So I figured out both parts of the hint, but I still have no clue how to make these hints help me here. Do I have to show that

    inf {[itex]\int[/itex][itex]_{B_{n}\cap E}[/itex]f d[itex]\mu[/itex], E[itex]\in[/itex]M, [itex]\mu (B_{n}\cap E)[/itex]≥α-δ} > 0 for each n? And then as n-> infinity we have the desired result? I'm still not sure how to prove this case though...
     
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