# Difference between Riemann-Stieltjes and Riemann Integral

dpa
Hi all,

## Homework Statement

Is the difference between riemann stieltjes integral and riemann integral that in riemann integral, the intervals are of equal length and in riemann stieltjes, the partitions are defined by the integrator function?

If not so what exactly is it that integrator function defines?

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Thank You

Homework Helper
2022 Award
Hi all,

## Homework Statement

Is the difference between riemann stieltjes integral and riemann integral that in riemann integral, the intervals are of equal length and in riemann stieltjes, the partitions are defined by the integrator function?

There is nothing in the definition of the Riemann integral which requires equal intervals. A Riemann sum for $f$ on $[a,b]$ is
$$\sum_{i=1}^{n} f(\xi_i)(x_i - x_{i-1})$$
where $x_{i-1} \leq \xi_i \leq x_i$, $x_0 = a$ and $x_n = b$. The corresponding expression for the Riemann-Stieltjes integral with integrator $g$ is
$$\sum_{i=1}^{n} f(\xi_i)(g(x_i) - g(x_{i-1}))$$
Thus the Riemann integral is the special case of the Riemann-Stieltjes integral where $g(x) = x$.

dpa
So, what exactly is it when people refer to "density" or similar notions when they discuss about Riemann Stieltjes integral. Is it how how fast alpha(x_i) grows? in the interval?