# How to prove Riemann-Stieltjes Integrals?

I have a Problem to prove Riemann-Stieltjes Integrals, following this :

1. If $M\subseteq[a,b]$ and $f$ and $g$ are functions with domain $M$ such that $f$ is $g-$ integrable over $M$, and there exists left(right) extension $f^*$ and $g^*$ to $[a,b]$, respectively, then $f^*$ is $g^*-$ integrable on $[a,b]$ and $$\int_{a}^{b} f^* dg^* = \int_M f dg$$

2. Suppose that $F$ and $G$ are functions with domain including $[a,b]$ such that

• $F$ is $G-$integrable on $[a,b]$
• $\overline{M}\subseteq[a,b]$ and $a,b \in M$
• if $z$ belong to $[a,b]-M$ and $\epsilon$ is a positive number, then there is an open interval $s$ containing $z$ such that $|F(x)-F(z)||G(v)-G(u)|<\epsilon$ where each of $u,v,$ and $x$ is in $s\cap[a,b],u<z<v,$ and $u\leq x\leq v$.

Then $F$ is $G$-integrable on $M$, and $\int_{a}^{b} f^* dg^* \ dx = \int_M f dg$

And I have problems to prove that and I have problems requests to a paper :

Robert M. McCleod, The generalized Riemann integral, Carus Mathematical Monographs, no. 20, The Mathematical Association of America, 1980.

Thank you very much.

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What do you mean by left(right) extension $f^*$? The extension needs to satisfy certain properties, otherwise there are clearly counter examples. So for example, do you mean continuous extension, or something else? – mez May 12 '13 at 11:27
$f$ is a function with domain $M$ $[a, b]$. By $f$ we mean a function such that (a) $f(x) = f(x)$ for each $x \in M$, and (b) if $x \in [a; b] - M$ and G is a gap containing x, then f(x) is equal to a quasi-end value of f with respect to G. It is understood that when there is more than one choice for f(x) then only one choice is made and is the same for each value in $G$. $f$ will be known as an extension of $f$ to $[a, b]$. If quasi-left end values are used consistently for each gap, then $f$ is known as a left extension of f on $[a,b]$. – Hirwanto May 12 '13 at 16:56
@mezhang ,, you can see more detail about definition $f*$ at paper with title "A STUDY OF A STIELTJES INTEGRAL DEFINED ON ARBITRARY NUMBER SETS" , and you can download paper at projecteuclid.org/… – Hirwanto May 12 '13 at 16:58