Denote by $R(g)$ the set of bounded real functions which are Riemann-Stieltjes integrable with respect to a given monotonically nondecreasing function $g$ on a given interval.

The Riemann-Stieltjes integral is a generalisation of the Riemann integral, and both have similar properties; N.B. however the items 5, 7 and 9.

1. If $f_1,\, f_2 \in R(g)$ on $[a,\,b]$ , then also $f_1\!+\!f_2,\,cf_1 \in R(g)$ on $[a,\,b]$ and
   $\int_a^b(f_1\!+\!f_2)dg = \int_a^bf_1\,dg+\int_a^bf_2\,dg,\;\; \int_a^bcf_1\,dg = c\int_a^bf_1\,dg$ .
2. If $f_1,\, f_2 \in R(g)$ on $[a,\,b]$ , then also $f_1f_2 \in R(g)$ on $[a,\,b]$ .
3. If $f_1,\,f_2 \in R(g)$ on $[a,\,b]$ and $\displaystyle\inf_{x\in[a,b]}|f_2(x)| > 0$ , 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
   $\int_a^bf_1\,dg \le \int_a^bf_2\,dg$ .
5. If $f \in R(g)$ on $[a,\,b]$ , and $V_g$ is the total variation of $g$ on $[a,\,b]$ , then
   $\left|\int_a^bfdg\right| \le$ $\displaystyle\sup_{x\in[a,b]}f(x)\cdot V_g$ .
6. If $f \in R(g)$ on $[a,\,b]$ , then also $|f| \in R(g)$ on $[a,\,b]$ and
   $\left|\int_a^bf\,dg\right| \le \int_a^b|f|\,dg$ .
7. If $f \in R(g)$ and $m \le f(x) \le M$ on $[a,\,b]$ , then
   $m[g(b)-g(a)] \le \int_a^bf\,dg \le M[g(b)-g(a)]$ .
8. If $f \in R(g)$ on $[a,\,b]$ and on $[b,\,c]$ , then also $f\in R(g)$ on $[a,\,c]$ and
   $\int_a^cf\,dg = \int_a^bf\,dg+\int_b^cf\,dg$ .
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
   $\int_a^bf\,d(g_1\!+\!g_2) = \int_a^bf\,dg_1+\int_a^bf\,dg_2$ .