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

1. Sep 12, 2010

### Boot20

1. The problem statement, all variables and given/known data

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

2. The attempt at a solution

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

Last edited: Sep 12, 2010