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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 functionMathworldPlanetmath 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  f1,f2∈R(g)  on  [a,b],  then also  f1+f2,cf1∈R(g) on  [a,b]  and
    ∫ba(f1+f2)𝑑g=∫baf1𝑑g+∫baf2𝑑g,∫bacf1𝑑g=c∫baf1𝑑g.

  2. 2.

    If  f1,f2∈R(g)  on  [a,b], then also  f1f2∈R(g) on  [a,b].

  3. 3.

    If  f1,f2∈R(g)  on  [a,b]  and  infx∈[a,b]|f2(x)|>0,  then also  f1f2∈R(g)  on  [a,b].

  4. 4.

    If  f1,f2∈R(g)  and  f1≤f2  on  [a,b],  then
    ∫baf1𝑑g≤∫baf2𝑑g.

  5. 5.

    If  f∈R(g)  on  [a,b],  and Vg is the total variationDlmfMathworldPlanetmathPlanetmath of g on  [a,b],  then
    |∫baf𝑑g|≤ supx∈[a,b]f(x)⋅Vg.

  6. 6.

    If  f∈R(g)  on  [a,b],  then also  |f|∈R(g) on  [a,b]  and
    |∫baf𝑑g|≤∫ba|f|𝑑g.

  7. 7.

    If  f∈R(g)  and  m≤f(x)≤M  on  [a,b],  then
    m[g(b)-g(a)]≤∫baf𝑑g≤M[g(b)-g(a)].

  8. 8.

    If  f∈R(g)  on  [a,b]  and on  [b,c],  then also   f∈R(g)  on  [a,c]  and
    ∫caf𝑑g=∫baf𝑑g+∫cbf𝑑g.

  9. 9.

    If  f∈R(g1),R(g2)  on  [a,b],  then  f∈R(g1+g2)  on the same interval and
    ∫bafd(g1+g2)=∫baf𝑑g1+∫baf𝑑g2.

  10. 10.

    If  f∈R(g)  on  [a,b],  then  g∈R(f)  on the same interval and one can integrate by parts:
    ∫baf𝑑g=f(b)g(b)-f(a)g(a)-∫bag𝑑f.

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