Related: When is it that $\int f d(g+h) \neq \int f dg + \int f dh$?
In this context, I write "integration" to mean the Riemann-Stieltjes integeation
Let $g:[a,b]\rightarrow \mathbb{R}$ be of bounded variation.
Let's define $\alpha(x)=V_a^x(g)$ and $g_1(x)= V_a^x (g) - g(x)$ and $g_2(x)= V_a^x(g) + g(x)$. ($V_a^x$ means the total variation) Then, these three functions are all monotonically increasing.
Let $f:[a,b] \rightarrow \mathbb{R}$ be a function.
If $f$ is integrable with respect to both $g_1$ and $g_2$, then obviously $f$ is integrable with respect to $\alpha$. (It is because $\alpha$ is the half the sum of $g_1$ and $g_2$)
I wonder whether the converse also holds.
That is,
If $f$ is integrable with respect to $\alpha$, would $f$ be integrable with respect to both $g_1$ and $g_2$?
Smirnov wrote in his text "A course of higher mathematics" that it is indeed true! But without proof. How do I prove this?
Fix $\epsilon>0$.
Let $U(P,f,\alpha)$ denote the upper sum and $L(P,f,\alpha)$ denote the lower sum and $m_i$ and $M_i$ be the infimum and supremum of $f$ on a subinterval $[x_{i-1}, x_i]$ where $P=\{x_0,...,x_n\}$ is a partition of $[a,b]$. So that $U(P,f,\alpha) = \sum M_i (\alpha(x_i) - \alpha(x_{i-1})$ and $L(P,f,\alpha) = \sum m_i (\alpha(x_i) - \alpha(x_{i-1})$.
Since $f$ is integrable along $\alpha$, there exists a partition $P$ of $[a,b]$ such that $U(P,f,\alpha) - L(P,f,\alpha) <\epsilon$.
Hence, $U(P,f,g_1 ) - L(P,f,g_1) = \sum (M_i - m_i) (g_1(x_i) - g_1(x_{i-1})) \leq \sum (M_i - m_i )(2 V_a^x(g)) = 2 ( U(P,f,\alpha) - L(P,f,\alpha)) < 2 \epsilon$.
Analogously, $f$ is integrable along $g_2$.