calculus - $alpha:(0, +infty) to (0, +infty)$ continuous and increasing, $alpha(1) = 1$. Show $int_1^x dalpha/alpha = log alpha(x)$

Let $\alpha:(0, +\infty) \to (0, +\infty)$ continuous and increasing,
with $\alpha(1) = 1$. Show that $\int_1^x d\alpha/\alpha = \log
\alpha(x)$ (don't suppose $\alpha$ differentiable)

Well, the change of variable theorem for Stieltjes integration says:

Let $f:[c,d]\to\mathbb{R}$ continuous, $\alpha:[c,d]\to\mathbb{R}$ of
bounded variation (rectifiable path) and $\phi:[a,b]\to[c,d]$ an

homeomorphism. Then:

$$\int_c^d f(t) d\alpha = \int_a^b f\circ\phi(s)d(\alpha\circ\phi)),
\mbox{ if $\phi$ is increasing } $$

$$\int_c^d f(t) d\alpha = -\int_a^b f\circ\phi(s)d(\alpha\circ\phi)),
\mbox{ if $\phi$ is decreasing } $$

I think I'm supposed to find the inverse homeomorphism from $\alpha$ (since its continuous and injective), in a way that $\alpha\circ\phi = 1$, so the integral becomes a Riemann integral. I should then evaluate $\int_{\phi(c)}^{\phi(d)} f\circ \phi(s) ds$ where $f: t\to \frac{1}{t}$. The integral now depends on $\phi(s)$, but how to Riemann integrate it?

Answer

Here is a simple change of variables result:

Claim. Let $f : [c, d] \to \mathbb{R}$ be Riemann integrable and $\alpha : [a, b] \to [c, d]$ be continuous and monotone increasing. Then

$$ \int_{a}^{b} f(\alpha(x)) \, d\alpha(x) = \int_{\alpha(a)}^{\alpha(b)} f(y) \, dy. $$

Proof. For each $\epsilon > 0$, pick $\delta > 0$ such that $|\alpha(x) - \alpha(y)| < \epsilon$ whenever $|x-y| < \delta$. Then for any partition $\Pi = \{a = x_0 < \cdots < x_n = b\}$ with $\|\Pi\| < \delta$ and for any $i = 1, \cdots, n$, we have

$$ m_i [\alpha(x_i) - \alpha(x_{i-1})] \leq \int_{x_{i-1}}^{x_i} f(\alpha(x)) \, d\alpha(x) \leq M_i [\alpha(x_i) - \alpha(x_{i-1})] $$

where

\begin{align*}
m_i &= \inf \{ f(y) : y \in [\alpha(x_{i-1}),\alpha(x_i)] \} \\
M_i &= \sup \{ f(y) : y \in [\alpha(x_{i-1}),\alpha(x_i)] \}.
\end{align*}

This tells that, with $\alpha(\Pi) = \{ \alpha(x_0), \cdots, \alpha(x_n)\}$ we have $\|\alpha(\Pi)\| < \epsilon$ and

$$ L(f, \alpha(\Pi)) \leq \int_{a}^{b} f(\alpha(x)) \, d\alpha(x) \leq U(f, \alpha(\Pi)), $$

where $L(f, \cdot)$ and $U(f, \cdot)$ are the lower Riemann sum and the upper Riemann sum, respectively. Therefore the desired identity follows by taking $\epsilon \to 0$.