# Stieltjes integral

1. May 3, 2006

### r4nd0m

Can you give me a simple real-life problem, where you need to use Stieltjes integral and can you show how you proceed in solving this kind of problems?

2. May 4, 2006

### HallsofIvy

Staff Emeritus
I'm not certain what you consider "real life"! I suppose it wouldn't help for me to point out that in "real life" you have to take a calculus test.

The Stieljes integral differs from the ordinary Riemann integral in that, after we have divided the interval (a to b, say) into n intervals with endpoints xi, xi+1, instead of defining $Delta xi to be simply xi+1- xi, that is, the length of the interval, we define it to be $\alpha(x_{i+1})- \alpha(x_i)$ where $\alpha(x)$ can be any increasing function. Taking the "Riemann sums" as usual then and taking the limit as the number of intervals goes to infinity results in the Stieltjes integral $\int f(x)d\alpha$ rather than the Riemann integral $\int f(x)dx$. Of course if $\alpha(x)$ happens to be differentiable then it is easy to see that $$\int f(x)d\alpha= \int f(x)\alpha'(x)dx$$ One common application is this: let $\alpha(x)$ be the "step" function (f(x)= 0 for 0<= x< 1, f(x)= 1 for 1<= x< 2, etc.). Then the sum $$\Sum_{n=0}^\k f(n)$$ can be written as the Stieltjes integral [tex]\int_0^{n+1} f(x)d\alpha$
allowing one to combine the theory of sums with integrals.

In particular