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I have an assignment where I have been given the following:

Given a measure space ($\mathbb{R}$, $B$, $\mu_F$) where $B$ is a Borel $\sigma$-field, and $\mu_F$ is the Lebesgue-Stieltjes measure generated from $F(x) = (1 - e^{-x})I(x\ge0)$.

I have found that

$\mu_F((a, b])$ = $\int_{(a, b]}e^{-x}I(x\ge0)d\mu(x)$,

where $\mu$ is the Lebesgue measure in $\mathbb{R}$ and I need now to show that for any measurable function $X$ in $(\mathbb{R}, B)$ with $X \ge 0$,

$\int X(x)d\mu_F(x) = \int X(x)e^{-x}I(x\ge0)d\mu(x)$

and have been given the hint to use a sequence of simple functions to approximate $X$.

Can someone please give me a clue as to how to even start this?

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  • $\begingroup$ Start by using the hint. $\endgroup$
    – user940
    Sep 16, 2016 at 21:45
  • $\begingroup$ @ByronSchmuland That's part of the problem, I have no idea how I would start on that. $\endgroup$
    – Aggie Kidd
    Sep 16, 2016 at 21:47
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    $\begingroup$ Can you show that $\mu_F(G)=\int_G e^{-x}I(x\ge 0)\,d\mu(x)$ for each open set $G$? $\endgroup$ Sep 16, 2016 at 21:53
  • $\begingroup$ @JohnDawkins I did this as well, but didn't realize until just now how this might be helpful. $\endgroup$
    – Aggie Kidd
    Sep 17, 2016 at 15:00

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