Riemann-Stieltjies Integrals

We assume throughout this section that F is an increasing function on a closed interval [a,b]. To avoid trivialities we assume F(a)<F(b). All left-hand and right-hand limits exist...We use the notation

No F (a confusing choice of variable) is a function that determines the size of intervals. In a Riemann integral
∫f dx
the interval [a,b] has size b-a
In a Riemann-Stieltjies integral
∫f dF
the interval [a,b] has size F(b+)- F(a-)

and of course when F(x)=x the Riemann-Stieltjies integral reduces to the Riemann integral

This is helpful in many ways.
-We can take sums as a type of integral and unify sums and integrals
-We can have impulse function like the Dirac delta function which concentrate a change to a single point.

No F (a confusing choice of variable) is a function that determines the size of intervals. In a Riemann integral
∫f dx
the interval [a,b] has size b-a
In a Riemann-Stieltjies integral
∫f dF
the interval [a,b] has size F(b+)- F(a-)

and of course when F(x)=x the Riemann-Stieltjies integral reduces to the Riemann integral

This is helpful in many ways.
-We can take sums as a type of integral and unify sums and integrals
-We can have impulse function like the Dirac delta function which concentrate a change to a single point.

Thanks a lot...is it ok if you give an example or something? So I can understand the difference better (between usual Reimann integrals and Reimann-Stieltjes Integrals)?

Where [itex] u_1(x) [/itex] is the uniform distribution on [0,0.5) and [itex] u_2(x) [/itex] is the uniform distribution on (0.5, 1] and the integrals are Riemann integrals.

It would be convenient to define a single distribution function for X and write [itex] \bar{f(x)} [/itex] as a single integral (even if the practical computation of that integral amounted to the work above). However, a Riemann integral can't handle the "point mass" probability at X = 0.5 because, in a manner of speaking, it sits on a rectangle whose base has zero length.

From the viewpoint of probability theory, a Riemann-Stieljes integral can be regarded as way of defining a new form of integration that handles such "point masses". ( You can define a nondecresasing function [itex] F(x) [/itex] which has a jump of size 0.3 at x = 0.5 )