properties of Riemann–Stieltjes integral

Denote by $R(g)$ the set of bounded real functions which are http://planetmath.org/node/3187Riemann–Stieltjes integrable with respect to a given monotonically nondecreasing function $g$ on  a given interval.

The http://planetmath.org/node/3187Riemann–Stieltjes integral is a generalisation of the Riemann integral, and both have properties; N.B. however the items 5, 7 and 9.

1. If  $f_1,f_2\in R(g)$  on  $[a,b]$,  then also  $f_1+f_2,c{f}_1\in R(g)$ on  $[a,b]$  and
   ${\int }_a^b(f_1+f_2)𝑑g={\int }_a^b{f}_1𝑑g+{\int }_a^b{f}_2𝑑g,{\int }_a^{b}c{f}_1𝑑g=c{\int }_a^b{f}_1𝑑g$.

2. If  $f_1,f_2\in R(g)$  on  $[a,b]$, then also  $f_1{f}_2\in R(g)$ on  $[a,b]$.

3. If  $f_1,f_2\in R(g)$  on  $[a,b]$  and  $\underset{x\in [a,b]}{inf}|f_2(x)|>0$,  then also $\frac{f_1}{f_2}\in R(g)$  on  $[a,b]$.

4. If  $f_1,f_2\in R(g)$  and  $f_1\le f_2$  on  $[a,b]$,  then
   ${\int }_a^b{f}_1𝑑g\le {\int }_a^b{f}_2𝑑g$.

5. If  $f\in R(g)$  on  $[a,b]$,  and $V_g$ is the total variation of $g$ on  $[a,b]$,  then
   $\left|{\int }_a^{b}f𝑑g\right|\le$ $\underset{x\in [a,b]}{sup}f(x)\cdot V_g$.

6. If  $f\in R(g)$  on  $[a,b]$,  then also  $|f|\in R(g)$ on  $[a,b]$  and
   $\left|{\int }_a^{b}f𝑑g\right|\le {\int }_a^b|f|𝑑g$.

7. If  $f\in R(g)$  and  $m\le f(x)\le M$  on  $[a,b]$,  then
   $m[g(b)-g(a)]\le {\int }_a^{b}f𝑑g\le M[g(b)-g(a)]$.

8. If  $f\in R(g)$  on  $[a,b]$  and on  $[b,c]$,  then also   $f\in R(g)$  on  $[a,c]$  and
   ${\int }_a^{c}f𝑑g={\int }_a^{b}f𝑑g+{\int }_b^{c}f𝑑g$.

9. If  $f\in R(g_1),R(g_2)$  on  $[a,b]$,  then  $f\in R(g_1+g_2)$  on the same interval and
   ${\int }_a^{b}fd(g_1+g_2)={\int }_a^{b}f𝑑g_1+{\int }_a^{b}f𝑑g_2$.

10. If  $f\in R(g)$  on  $[a,b]$,  then  $g\in R(f)$  on the same interval and one can integrate by parts:
    ${\int }_a^{b}f𝑑g=f(b)g(b)-f(a)g(a)-{\int }_a^{b}g𝑑f$.