# 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. 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
$\int_{a}^{b}(f_{1}\!+\!f_{2})dg=\int_{a}^{b}f_{1}\,dg+\int_{a}^{b}f_{2}\,dg,\;% \;\int_{a}^{b}cf_{1}\,dg=c\int_{a}^{b}f_{1}\,dg$.

2. 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. 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. 4.

If  $f_{1},\,f_{2}\in R(g)$  and  $f_{1}\leq f_{2}$  on  $[a,\,b]$,  then
$\int_{a}^{b}f_{1}\,dg\leq\int_{a}^{b}f_{2}\,dg$.

5. 5.

If  $f\in R(g)$  on  $[a,\,b]$,  and $V_{g}$ is the total variation    of $g$ on  $[a,\,b]$,  then
$\left|\int_{a}^{b}fdg\right|\leq$ $\displaystyle\sup_{x\in[a,b]}f(x)\cdot V_{g}$.

6. 6.

If  $f\in R(g)$  on  $[a,\,b]$,  then also  $|f|\in R(g)$ on  $[a,\,b]$  and
$\left|\int_{a}^{b}f\,dg\right|\leq\int_{a}^{b}|f|\,dg$.

7. 7.

If  $f\in R(g)$  and  $m\leq f(x)\leq M$  on  $[a,\,b]$,  then
$m[g(b)-g(a)]\leq\int_{a}^{b}f\,dg\leq M[g(b)-g(a)]$.

8. 8.

If  $f\in R(g)$  on  $[a,\,b]$  and on  $[b,\,c]$,  then also   $f\in R(g)$  on  $[a,\,c]$  and
$\int_{a}^{c}f\,dg=\int_{a}^{b}f\,dg+\int_{b}^{c}f\,dg$.

9. 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
$\int_{a}^{b}f\,d(g_{1}\!+\!g_{2})=\int_{a}^{b}f\,dg_{1}+\int_{a}^{b}f\,dg_{2}$.

10. 10.

If  $f\in R(g)$  on  $[a,\,b]$,  then  $g\in R(f)$  on the same interval and one can integrate by parts:
$\int_{a}^{b}f\,dg\;=\;f(b)g(b)\!-\!f(a)g(a)\!-\int_{a}^{b}g\,df$.

Title properties of Riemann–Stieltjes integral PropertiesOfRiemannStieltjesIntegral 2013-03-22 18:54:59 2013-03-22 18:54:59 pahio (2872) pahio (2872) 12 pahio (2872) Topic msc 26A42 properties of Riemann-Stieltjes integral ProductAndQuotientOfFunctionsSum FactsAboutRiemannStieltjesIntegral