What is the difference between Riemann and Riemann-Stieltjes integrals?

I'm quite confused, what is the difference between these two integrals (R and RS)? It seems that RS is closer to Lebesgue in its treatment of discontinuities, but otherwise I don't understand. If someone could give an example of a function for which they were different, it would be very beneficial.

Thanks.

It seems to me that you are integrating relative to a $dg(x)$, rather than $dx$. For example, the if $g(x)$ is $0$ for negative $x$ and $1$ for positive $x$, then then $\int_{-1}^{1} f(x)dg(x)$ is $f(0)$ if $f$ is continuous.

If $g(x)=x$, the Riemann-Stieltjes integral is just the Riemann integral.

If $g(x)$ is continuously differentiable, then the RS-integral $\int_{a}^{b} f(x)dg(x)$ is the same as the Riemann integral $\int_a^b f(x)g'(x) dx$.

The differences then are the cases where $g(x)$ is not continuously differentiable. For example, if $g(x)$ is the step function above, then $dg(x)$ is "like" the Dirac delta function.

It's a beginning of a way of thinking of integrals as operators.

• For the case with Dirac delta function, doesn't the RS integral reduce to R = $\int_{0}^{1}f(x)dx$? Jan 25 '12 at 4:06
• Nope, because if $g$ is the step function, and you partition the interval $-1=x_0<x_1<..<x_n=1$, the sum $\sum_i f(c_i)(g(x_{i+1})-g(x_i))$ is zero at every $i$ except where $x_i\leq 0 < x_{i+1}$, and, in that term, $g(x_{i+1})-g(x_i)$ is $1$, so the value of the sum is $f(c_i)$ where $c_i$ is in a neighborhood of $0$. That means that the limit will be $f(0)$ if $f$ is continuous at $0$. If $g(x)=0$ for $x\leq 0$ and $g(x)=x$ for $x>0$ then $\int_{-1}^1 f(x)dg(x)=\int_{0}{1} f(x)fx$. Jan 25 '12 at 4:31
• did you mean $\int_{0}^{1}f(x)dx$? Jan 25 '12 at 6:06
• Yes, got a typo in there. Jan 25 '12 at 7:53

i think the difference is that , in riemann integral we used function f(x) but in riemann - stieljes integral we use one additional function i.e. alpha of x which is monotonically increasing and bounded function.

Another way of saying this is that in the "Riemann-Stieltjes" we compute the "size" of each interval by $\alpha(x_{n+1})- \alpha(x_n)$ where $\alpha$ is an increasing function. If $\alpha$ is a differentiable function, then the Riemann-Stieltjes integral $\int f(x)d\alpha$ is the same as the Riemann integral $\int f(x)\frac{d\alpha}{dx}dx$. However, if $\alpha$ is not differentiable (and it does not even have to be continuous) the Riemann-Stieljes integral will exist while the Riemann integral does not. A popular use of the Riemann-Stieljes integral is to take $\alpha$ to be a step function so that a sum can be written as an integral.