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Use the Carathéodory extension theorem to find the Lebesgue-Stieljes measure of F(x)

  1. Sep 12, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex](R, \mathcal{B}, \mu_F)[/tex] be a measure space, where [tex]\mathcal{B}[/tex] is the Borel [tex]\sigma[/tex]-filed and [tex]\mu_F[/tex] is the Lebesgue-Stieljes measure generated from
    [tex]F(x) = \sum^\infty_{n=1}2^{-n}I(x \ge n^{-1}) + (e^{-1} - e^{-x})I(x \ge 1)[/tex]
    Use the uniqueness of measure extension in the Carathéodory extension theorem to show
    [tex]\mu_(B) = \sum^\infty_{n=1}2^{-n}I(n^{-1} \in B) +
    \int_B e^{-x}I(x \ge 1)d\lambda(x)[/tex]
    for any [tex]B \in \mathcal{B}[/tex], where [tex]\lambda[/tex] is the Lebesgue measure.

    2. The attempt at a solution

    I tried to use the Caratheodory extension theorem,
    [tex]
    \mu_F(B)
    &= \inf\left\{ \sum^{\infty}_{i=1}\mu_F(B_i): B_i \in \mathcal{B}_0, B \subset \cup^\infty_{i=1}B_i \right\}
    [/tex]
    and separating the cover of B into a cover of [0,1] and a cover of [tex](1,\infty)[/tex]. I can do the first cover just fine but the second cover baffles me. How do I prove that it is equal to [tex]\int_B e^{-x}I(x \ge 1)d\lambda(x)[/tex]
     
    Last edited: Sep 12, 2010
  2. jcsd
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