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Integration using a cumulative distribution function

  1. Jul 29, 2011 #1
    Hi all,
    I'm really banging my head on this problem:
    Let f be a real-valued measurable function on the measure space [tex](X,\mathcal{M},\mu).[/tex]
    Define
    [tex]\lambda_f(t)=\mu\{x:|f(x)|>t\}, t>0.[/tex]
    Show that if [itex]\phi[/itex] is a nonnegative Borel function defined on [0,infinity), then
    [tex]\int_0^{\infty}\phi(|f(x)|)d\mu=-\int_0^{\infty}\phi(t)d\lambda_f(t).[/tex]

    A hint is given, which is to look at
    [tex]\nu((a,b])=\lambda_f(b)-\lambda_f(a)=-\mu\{x:a<|f(x)|\leq b\},[/tex]
    and argue that it extends uniquely to a Borel measure.

    This is straightforward, I think, as closed intervals form a semi-ring and [itex]\nu[/itex] is a premeasure. I'm just not sure where to go from here. Any help would be greatly appreciated, thanks!
     
  2. jcsd
  3. Jul 30, 2011 #2

    mathman

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    Science Advisor
    Gold Member

    It looks like it could be solved by using the elementary approach to defining Lebesgue integrals - dividing the y (or f) axis into strips and taking a limit as the strip widths go to 0.

    The strips would correspond to (a,b] in the hint.
     
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