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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!

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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 $f \in {\cal R}_{[a,b]}(\alpha)$.

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