# Showing Riemann-Stieltjes Integrability

I am doing an independent study in Measure Theory, and I am using Measure and Integral by Wheeden and Zygmund. In chapter 2, "Functions of Bounded Variation and the Riemann-Stieltjes Integral", I am working on problem 16, but I am having a bit of trouble determining how to approach it. Here is the problem statement.

Suppose that $$\phi$$ is of bounded variation on $$[a,b]$$ and that $$f$$ is bounded and continuous except for a finite number of jump discontinuities in $$[a,b]$$. If $$\phi$$ is continuous at every point of discontinuity of $$f$$, show that $$\int_{a}^{b}fd \phi$$ exists.

I would appreciate any pointers on how to approach the proof. Thanks!

Since $$\phi$$ is BV, we can write $$\phi=\alpha-\beta$$ where $$\alpha, \beta$$ are non decreasing and are both continuous wherever $$\phi$$ is.
It is sufficient to show that $$\int_a^b f d \alpha$$ exists. Let $$d_k$$ be the ordered points of discontinuities in $$[a,b]$$.
Choose $$\epsilon>0$$. Let $$B$$ be a bound on $$f$$.
Since $$\alpha$$ is continuous at each $$d_k$$, there is some $$\eta>0$$ such that $$\sum_k \alpha(d_k+\eta)-\alpha(d_k-\eta) < \epsilon$$. Note that we can reduce the value of $$\eta$$ and the inequality remains true. Choose $$\eta$$ small enough so that the intervals $$U_k=(d_k+\eta,d_k-\eta)$$ do not overlap. Note that $$[a,b] \setminus \cup_k U_k$$ is a collection of intervals $$I_1,...,I_m$$ on which $$f$$ is continuous. Since $$f \in {\cal R}_{I_j}(\alpha)$$, there are partitions $$P_j$$ such that $$\sum_j U_{I_j} (f,\alpha,P_j) - L_{I_j} (f,\alpha,P_j) < \epsilon$$.
Now combine the partitions $$P_j$$ to get a partition $$P$$ of $$[a,b]$$, then $$U_{[a,b]} (f,\alpha,P_j) - L_{[a,b]} (f,\alpha,P_j) < \epsilon + 2B\epsilon$$, and since $$\epsilon>0$$ was arbitrary, we see that $$f \in {\cal R}_{[a,b]}(\alpha)$$.