Show that if $f \in R_\alpha$,and g increasing and continuous then $ f(g(x)) \in R_{\alpha(g(x))}$ Let $f \in R_\alpha[a,b]$ and $g:[c,d] \rightarrow \mathbb{R}$ continuous and strictly increasing, such that $g(c) = a$ and $g(d) = b$. Prove that $f(g(x)) \in R_{\alpha(g(x))}$.
In calculus I remember to prove something similar for a $f$ Riemann integrable function and $g$ a continuos strictly increasing function then $f(g(x))$ was a Riemann integrable function.
I almost 99.99% convince that this is the case for the Riemann Stieljets integral, becouse of the bijectivity of the function g, Playing with some functions make me think that $$\int_c^d f(g(x))d(\alpha(g(x))) = \int _c^d h(x)d(\beta(x))$$
Where $h(x) = f(g(x))$ and $\beta(x) = \alpha(g(x))$ and $g([c,d]) = [a,b]$
Not so sure about the last one tough.
Could you guys help me? 
 A: Since $f$ is RS-integrable with respect to $\alpha$, for any $\epsilon > 0$ there exists a partition $P_\epsilon$ of $[a,b]$ with points $ a = y_0 < y_1 < \ldots < y_m = b$ such that if $P$ is a refining partition with points  $ a = x_0 < x_1 < \ldots < x_n = b$ , then for any Riemann-Stieltjes sum 
$$S(P,f,\alpha) = \sum_{j=1}^n f(\xi_j)( \, \alpha(x_j) - \alpha(x_{j-1})\, )$$ 
where $\xi_j \in [x_{j-1}, x_j]$ are any intermediate points, we have 
$$\left|S(P,f,\alpha) - \int_a^b f \, d\alpha \right| < \epsilon$$
Since $g$ is strictly increasing and continuous there exists a continuous inverse $g^{-1}:[a,b] \to [c,d]$ such that $g^{-1}(a) = c$ and $g^{-1}(b) = d$
This gives a partition $P'_\epsilon = g^{-1}(P_\epsilon) = \{g^{-1}(y_0), g^{-1}(y_1), \ldots , g^{-1}(y_m)\}$ such that, for any $P$ that refines $P_\epsilon$, the partition $P' =g^{-1}(P)=  \{g^{-1}(x_0), g^{-1}(x_1), \ldots , g^{-1}(x_n)\} $ is a refinement of $g^{-1}(P_\epsilon)$. There is a one-to-one correspondence between all partitions $P$ of $[a,b]$ and all partitions $P'$ through the mapping $g^{-1}$.
Also by monotonicity, for any intermediate point $\xi_j \in [x_{j-1},x_j]$ the corresponding point is $g^{-1}(\xi_j) \in [g^{-1}(x_{j-1}), g^{-1}(x_j)]$.
Note that $f(\xi_j) = f(g(g^{-1}(\xi_j)))$ and  $\alpha(x_j) = \alpha(g(g^{-1}(x_j)))$,
Thus,
$$\left|S(P',f \circ g,\alpha \circ g) - \int_a^b f \, d\alpha \right|=\left|S(P,f,\alpha) - \int_a^b f \, d\alpha \right| < \epsilon$$
This proves that the composition $f \circ g$ is Riemann-Stieltjes integrable with respect to the composition $\alpha \circ g$
