Currently, I am stuck at Problem 57 in Ch 6 of Real Analysis, 4th Ed by Royden. In the previous problem, I have proven that
Let $g$ be strictly increasing and absolutely continuous on $[a, b]$. Then for any non-negative integrable $f$ over $[c,d]$, $$ \int^d _c f(y) dy = \int ^b _a f(g(x))g'(x) dx \tag{*} $$
The problem I am working on asked about
Is the formula (*) true if $g$ is just increasing, not strictly increasing?
I know that the statement is true as it is stated in other literature. However, I cannot extend the result from strictly increasing to merely increasing $g$.
My attempts:
Define $g_n (x) = g(x) + x/n$, where $g$ is increasing so $g_n$ is strictly increasing on $[a, b]$. Let $f$ be a non-negative integrable function on $[c, d] = [g(a), g(b)]$. Further define
$$
\alpha = \min\{c, c+a\} \quad \text{and} \quad \beta = \max \{ d, d + b\}
$$
and extend $f$ to $[\alpha, \beta]$ by setting $f = 0$ outside $[c, d]$. From (*), for all $n$,
\begin{align*} \int^{d+b/n} _{c + a/n} f(y) dy &= \int ^b _a f(g_n(x))g_n'(x) dx \\ \implies \lim _{n \to \infty} \int^{d+b/n} _{c + a/n} f(y) dy &= \lim _{n \to \infty} \int ^b _a f(g_n(x))g_n'(x) dx \\ &= \lim _{n \to \infty} \int ^b _a f\left(g(x) + \frac{x}{n}\right) \left[g'(x) + \frac{1}{n}\right] dx \end{align*}
I can show that the left hand side converges to $\int^d _c f(y) dy$. Note that $$ \int^{d+b/n} _{c + a/n} f(y) dy = \int^{\beta} _{\alpha} f(y) \cdot \chi_{[c+a/n, d + b/n]}(y) \, dy $$ Since $f \circ \chi_{[c+a/n, d + b/n]} \to f \circ \chi_{[c, d]}$ pointwise (or $ f \circ \chi_{(c ,d)}$), and $$ |f \circ \chi_{[c+a/n, d + b/n]}| \leq |f| \text{ for all $n$ on } [\alpha, \beta] $$ so by the Lebesgue Dominated Convergence Theorem, $$ \lim _{n \to \infty} \int^{d+b/n} _{c + a/n} f(y) dy = \int _{\alpha} ^\beta f(y) \chi_{[c, d]} (y) dy = \int _{d} ^c f(y) dy $$
The right hand side is where I am stuck at. Here is another post about a similar problem, which mentioned approximating $f$ by a continuous function, but I cannot see how using a continuous approximation helps. The best we know is that for every $\epsilon > 0$ there is a continuous function $h$ on $[\alpha, \beta]$ which $$ \int _\alpha ^\beta |f - h| < \epsilon $$ And applying (*) on it gives \begin{align*} \int _a ^b \left| f\left(g(x) + \frac{x}{n}\right) - h\left(g(x) + \frac{x}{n}\right) \right| \left[g'(x) + \frac{1}{n}\right] dx &< \epsilon \\ \implies \left| \int _a ^b \left[ f\left(g(x) + \frac{x}{n}\right) - h\left(g(x) + \frac{x}{n}\right) \right] \left[g'(x) + \frac{1}{n}\right] dx \right| &< \epsilon \\ \implies \int _a ^b f\left(g(x) + \frac{x}{n}\right) \left[g'(x) + \frac{1}{n}\right] dx &< \epsilon + \int _a ^b h\left(g(x) + \frac{x}{n}\right) \left[g'(x) + \frac{1}{n}\right] dx \end{align*} If somehow we can exchange limit on the right hand side, then as $h$ is continuous, $$ \lim _{n \to \infty} \int _a ^b f\left(g(x) + \frac{x}{n}\right) \left[g'(x) + \frac{1}{n}\right] dx \leq \epsilon + \int _a ^b h(g(x)) g'(x) dx $$ But the integral on the right is still far from the desired integral, which is $$ \int _a ^b f(g(x)) g'(x) dx $$ and we only have inequality instead of equality.
I have no idea how to proceed. I know that (*) can be proven without assuming $g$ is strictly increasing, but I would still like to know how to extend (*) to strictly increasing functions. Any advice would be appreciated.