Riemann-Stieltjes Integrals with Mult. Disc. Step Func. as Integrators

# Riemann-Stieltjes Integrals with Multiple Discontinuity Step Functions as Integrators

Recall from the Riemann-Stieltjes Integrals with Single Discontinuity Step Functions as Integrators page that if $f$ is a function on the interval $[a, b]$ and $\alpha$ is a step function with a single discontinuity, say $\alpha (x) = \left\{\begin{matrix} p & \mathrm{for \: all} \: x \in [a, c) \\ q & \mathrm{for} \: x = c\\ r & \mathrm{for \: all} \: x \in (c, b] \end{matrix}\right.$, then provided that one of $f$ or $\alpha$ is left continuous at $c$ AND one of $f$ or $\alpha$ is right continuous at $c$ then $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and:

(1)
\begin{align} \quad \int_a^b f(x) \: d\alpha(x) = f(c)[\alpha(c^+) - \alpha(c^-)] \end{align}

We will now extend this theorem for a step function with $n$ discontinuities. The proof can be established with induction but is rather cumbersome, so we will only state the result.

 Theorem 1: Let $f$ be a function on the interval $[a, b]$ and let $\alpha$ be a step function with jump discontinuities at $x_1, x_2, ..., x_n \in [a, b]$. Then, if at each $x_k$ for $k = 1, 2, ..., n$ we have that not both $f$ and $\alpha$ are discontinuous from the left and not both $f$ and $\alpha$ are discontinuous from the right, then $f$ is Riemann-Stietljes integrable with respect to $\alpha$ on $[a, b]$ and furthermore $\displaystyle{\int_a^b f(x) \: d\alpha(x) = \sum_{k=1}^{n} f(x_k)[\alpha(x_k^+) - \alpha(x_k^-)}]$. If for some $k \in \{1, 2, ..., n \}$ we have that $x_k = a$ then we denote $\alpha(x_k^+) - \alpha(x_k^-) = \alpha(a^+) - \alpha(a)$, and if $x_n = b$, and if for some $k \in \{1, 2, ..., n \}$ we have that $x_k = b$ then we denote $\alpha(x_k^+) - \alpha(x_k^-) = \alpha(b) - \alpha(b^-)$.