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 $f_{1}\!+\!f_{2},\,cf_{1}\in R(g)$ 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.

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 $\displaystyle\inf_{x\in[a,b]}|f_{2}(x)|>0$ , then also $\frac{f_{1}}{f_{2}}\in R(g)$ on $[a,\,b]$ .
4.

If $f_{1},\,f_{2}\in R(g)$ and $f_{1}\leq f_{2}$ on $[a,\,b]$ , then
$\int_{a}^{b}f_{1}\,dg\leq\int_{a}^{b}f_{2}\,dg$ .
5.

If $f\in R(g)$ on $[a,\,b]$ , and $V_{g}$ is the total variation of $g$ on $[a,\,b]$ , then
$\left|\int_{a}^{b}fdg\right|\leq$ $\displaystyle\sup_{x\in[a,b]}f(x)\cdot V_{g}$ .
6.

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

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.

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.

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
Canonical name	PropertiesOfRiemannStieltjesIntegral
Date of creation	2013-03-22 18:54:59
Last modified on	2013-03-22 18:54:59
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	12
Author	pahio (2872)
Entry type	Topic
Classification	msc 26A42
Synonym	properties of Riemann-Stieltjes integral
Related topic	ProductAndQuotientOfFunctionsSum
Related topic	FactsAboutRiemannStieltjesIntegral