Property of Riemann Stieltjes Integral

Question

I proved the following properties of the Riemann-Stieltjes integral without too much difficulty:

$$ \int_a^b cf \, d\phi = c \int_a^b f \, d\phi $$ $$ \int_a^b (f_1 + f_2) \, d\phi = \int_a^b f_1 \, d\phi + \int_a^b f_2 \, d\phi $$ $$ \int_a^b f \, d(\phi_1 + \phi_2) = \int_a^b f \, d\phi_1 + \int_a^b f \, d\phi_2 $$

assuming that the integrals in the RHS exist.

However, I'm having a hard time proving the following property:

$$ \int_a^b f \, d(\phi_1 \phi_2) = \int_a^b f \phi_1 \, d\phi_2 + \int_a^b f \phi_2 \, d\phi_1 $$

where $f$, $\phi_1$, and $\phi_2$ are continuous and of bounded variation on $[a,b]$. Can anyone give me some hint on how to approach this? Thanks!

Answer 1

For any tagged partition $P: a \leqslant \xi_1 \leqslant x_1 \leqslant \xi_2 \leqslant x_2 \leqslant \ldots \leqslant x_{n-1} \leqslant \xi_n \leqslant x_n$ the Riemann-Stieltjes sum can be decomposed as

$$S(P,f,\phi_1\phi_2) = \sum_{k=1}^n f(\xi_k)[\phi_1(x_k)\phi_2(x_k) - \phi_1(x_{k-1})\phi_2(x_{k-1})] \\= S(P,f\phi_1, \phi_2)+ S(P, f \phi_2, \phi_1) +S_C +S_D, $$

where $$S(P,f \phi_1, \phi_2) = \sum_{k=1}^n f(\xi_k)\phi_1(\xi_k)[\phi_2(x_k) - \phi_2(x_{k-1})] , \\ S(P, f \phi_2, \phi_1) = \sum_{k=1}^n f(\xi_k)\phi_2(\xi_k)[\phi_1(x_k) - \phi_1(x_{k-1})], \\ S_C = \sum_{k=1}^n f(\xi_k)[\phi_1(x_k)-\phi_1(\xi_k)][\phi_2(x_k) - \phi_2(x_{k-1})] , \\ S_D = \sum_{k=1}^n f(\xi_k)[\phi_2(x_{k-1})-\phi_2(\xi_k)][\phi_1(x_k) - \phi_1(x_{k-1})].$$ Hence,

$$\tag{1}\left|S(P,f,\phi_1\phi_2) - \left(\int_a^b f \phi_1 \, d\phi_2 + \int_a^b f \phi_2 \, d\phi_1 \right)\right| \leqslant \\ \left|S(P,f \phi_1, \phi_2) - \int_a^b f \phi_1 \, d\phi_2\right| \\ + \left|S(P,f \phi_2, \phi_1) - \int_a^b f \phi_2 \, d\phi_1\right| \\ + |S_C| + |S_D|.$$

For any $\epsilon >0$ there exists a partition $P_\epsilon$ such that for any refinement $P$, each of the terms on the right-hand side of (1) is smaller than $\epsilon/4.$ The first two terms can be made small in this way because $f \phi_1$ and $f \phi_2$ are Riemann-Stieltjes integrable with respect to $\phi_2$ and $\phi_1$. We have $S_C , S_D < \epsilon /4$ for sufficiently fine partitions because all functions are continuous and of bounded variation. For example, with $M = \sup_{x \in [a,b]} |f(x)|$ we have

$$|S_C| \leqslant \sum_{k=1}^n |f(\xi_k)||\phi_1(x_k)-\phi_1(\xi_k)||\phi_2(x_k) - \phi_2(x_{k-1})| \\ \leqslant M \sum_{k=1}^n |\phi_1(x_k)-\phi_1(\xi_k)||\phi_2(x_k) - \phi_2(x_{k-1})|. $$

Let $V_a^b(\phi_2)$ denote the variation of $\phi_2$ on $[a,b]$. Since $\phi_1$ is continuous there exists $\delta > 0$ such that if $|\xi_k - x_k| \leqslant |x_k - x_{k-1}| < \delta$ for all $k$ then $|\phi_1(x_k) - \phi_1(\xi_k)| < \epsilon/(4MV_a^b(\phi_2)), $ and

$$|S_C| < M \frac{\epsilon}{4M V_a^b(\phi_2)}\sum_{k=1}^n |\phi_2(x_k) - \phi_2(x_{k-1})| < \frac{\epsilon}{4}.$$

It follows that for every partition $P$ that refines $P_\epsilon$ we have

$$\left|S(P,f,\phi_1\phi_2) - \left(\int_a^b f \phi_1 \, d\phi_2 + \int_a^b f \phi_2 \, d\phi_1 \right)\right| < \epsilon.$$

Therefore,

$$\int_a^n f \, d(\phi_1 \phi_2) = \int_a^b f \phi_1 \, d\phi_2 + \int_a^b f \phi_2 \, d\phi_1 .$$