properties of Riemann–Stieltjes integral

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

The–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  f1,f2R(g)  on  [a,b],  then also  f1+f2,cf1R(g) on  [a,b]  and

  2. 2.

    If  f1,f2R(g)  on  [a,b], then also  f1f2R(g) on  [a,b].

  3. 3.

    If  f1,f2R(g)  on  [a,b]  and  infx[a,b]|f2(x)|>0,  then also  f1f2R(g)  on  [a,b].

  4. 4.

    If  f1,f2R(g)  and  f1f2  on  [a,b],  then

  5. 5.

    If  fR(g)  on  [a,b],  and Vg is the total variationDlmfMathworldPlanetmathPlanetmath of g on  [a,b],  then
    |abf𝑑g| supx[a,b]f(x)Vg.

  6. 6.

    If  fR(g)  on  [a,b],  then also  |f|R(g) on  [a,b]  and

  7. 7.

    If  fR(g)  and  mf(x)M  on  [a,b],  then

  8. 8.

    If  fR(g)  on  [a,b]  and on  [b,c],  then also   fR(g)  on  [a,c]  and

  9. 9.

    If  fR(g1),R(g2)  on  [a,b],  then  fR(g1+g2)  on the same interval and

  10. 10.

    If  fR(g)  on  [a,b],  then  gR(f)  on the same interval and one can integrate by parts:

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