Let $g,h:[a,b]\rightarrow \mathbb{R}$ be of bounded variation.
Let $f:[a,b]\rightarrow \mathbb{R}$ be a function which is Riemann-Stieltjes integrable along $g$ and $h$.
** Problem1**
When is it that $\int_a^b fd(g+h) \neq \int_a^b fdg + \int_a^b fdh$?
(Nevertheless, I have proven that if $f$ is continuous, the equality holds.)
Define $g_1(x)=1/2(V_a^x(g) + g(x))$ and $g_2(x)=1/2(V_a^x(g) - g(x))$. Then $g=g_1-g_2$ and let's call this the canonical decomposition of $g$.
(Note that it is the definition to say "$f$ is integrable along $g$ iff $f$ is integrable along $g_1$ and $g_2$".)
Let $h_1,h_2,(g+h)_1,(g+h)_2$ be the canonical decomposition of $h,g+h$.
Problem2
Since $V_a^x(g+h)\leq V_a^x (g)+V_a^x(h)$, even though $f$ is integrable along $g,h$, it does not gurantee that $f$ is integrable along $g+h$. When is $f$ integrable along $g+h$?
Thank you in advance.
I will use the result and notations I answered in: Necessary and sufficient condition for Integrability along bounded variation
It is sufficient to prove that $f$ is integrable along the total variation of $g+h$.
Since $U(P,f,V_a^x (g+h)) - L( P,f,V_a^x (g+h)) = \sum (M_i -m_i ) V_{x_{i-1}}^{x_i} (g+h) \leq \sum (M_i - m_i ) (V_{x_{i-1}}^{x_i} (g) + V_{x_{i-1}}^{x_i} (h))$, $f$ is integrable along $g+h$.