Integrator monotonically increasing ? R.S. Integral

[sihag]

Could someone explain why the R.S. Integral is defined for a monotonically increasing integrator? Can't we use a decreasing fuction anologously?

 

[HallsofIvy]

Yes, you could. It would just give the negative of the corresponding integral with an increasing integrator.

(For anyone who is wondering, "R.S" is the Riemann-Stieljes integral. It is defined exactly like the Riemann integral except that instead of measuring the size of each interval forming the base of a rectangle as $x_{i+1}- x_i$, we use $\alpha(x_{i+1})-\alpha(x_i)$ where $\alpha(x)$ can be an monotone increasing function of x. It is typically written $\int f(x)d\alpha(x)$. If $\alpha$ is a differentiable function, the Riemann-Stieljes integral is exactly the same as the Riemann integral $\int f(x) d\alpha/dx dx$. The interesting situation is when $\alpha$ is not differentiable. In particular, if $\alpha$ is the unit step function, and a and b are integers, then $\int_a^b f(x) d\alpha= f(a)+ f(a+1)+ \cdot\cdot\cdot+ f(b)$.)