(adsbygoogle = window.adsbygoogle || []).push({}); 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]

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

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