Let f and $\alpha$ be bounded functions on $[a,b]$ and suppose that $\int_{a}^{b} f(x) d\alpha(x)$ exists. Let $g: [c,d] \rightarrow [a,b]$ be strictly increasing continuous function with $g(c)=a $and $g(d)=b$.
Define $h(x) = f(g(x))$ and $\beta (x) = \alpha(g(x))$ .
My question is how would one show that the formula for substitution below is valid :
$$\int_{a}^{b} f(x) d\alpha(x) = \int_{c}^{d} h(x) d\beta(x)$$
Thank you!
Pick any partition $P=\{x_0, \ldots, x_n\}$ on $[a,b]$. It is easy to check that $g$ is surjective so that for each $x_i,$ there is $y_i$ such that $g(y_i)=x_i$, form a partition $Q$ with those $y_i$. If you can observe what happens to $U(Q, h, \beta)$ and $L(Q, h, \beta)$, it shall be fairly easy to check the integrability and then the equality.