Composition of absolutely continuous functions with monotonic condition

Question

$f$ is absolutely continuous on $\mathbb{R}$, and $g$ is absolutely continuous on $[a,b]$ and strictly monotone. Show $f \circ g$ is absolutely continuous.

I'm using the $\epsilon-\delta$ definition of absolute continuity (AC) and I have some doubts:

1. It is given that $f$ is absolutely continuous on $\mathbb{R}$ however the definition I'm using uses a closed and bounded interval. Can I say that since $f$ is absolutely continuous on $\mathbb{R}$, it is AC on any arbitrary $[c,d]$ on $\mathbb{R}$ and apply the definition? If not are there any equivalent definitions?

2. If I can do above, I'm having difficulty in choosing the $\epsilon-\delta$ for $f,g$ on $[c,d]$ and $[a,b]$ and using the monotonic condition of $g$.

3. One sufficient condition is $f$ being Lipschitz on $\mathbb{R}$ and $g$ being any AC function on $[a,b]$, not necessarily monotonic. Can that be used here somehow?

Answer 1

Assume

1. $g$ is absolutely continuous on $[a,b]$ and strictly increasing. Since it is continuous there is an interval $[c,d]$ so that $g:[a,b]\to [c,d]$.
2. Assume that $f:[c,d]\to \mathbb R$ is absolutely continuous. That means the composition $f\circ g:[a,b]\to \mathbb R$. (You won't need that $f$ is absolutely continuous on all of $(-\infty,\infty)$ which is a stronger statement.)
3. Now do your thing. Let $\epsilon=\epsilon_1>0$ and use the absolute continuity of $f$ to get your $\delta_1$ etc.
4. Now do it for $g$. Using $\epsilon_2=\delta_1$ find a $\delta_2$ etc.

I think you have the idea here. You want to end up with something like this:

$$\sum (b_i-a_i) <\delta_2 \implies \sum [g(b_i)-g(a_i)] < \delta_1$$

write $[c_i,d_i] = [g(a_i),g(b_i)]$ then

$$\sum (d_i-c_i) <\delta_1 \implies \sum |f(d_i)-f(c_i)| < \epsilon$$

so that

$$\sum (b_i-a_i) <\delta_2 \implies \sum |f\circ g(b_i)-f\circ g(a_i)| < \epsilon$$

That's the vague goal. Just write it up formally as a proof. Make it very clear where you are using the fact that $g$ is strictly increasing.

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Now there is a lot more to learn about compositions of absolutely continuous functions as this special case is very elementary. See S. Saks, Theory of the Integral, (1937), pp.286-289.

https://archive.org/details/theoryoftheinteg032192mbp/page/n9/mode/2up

It was known for a long time that the composition of two absolutely continuous functions need not be absolutely continuous, except in special cases like this.

Two famous Russian mathematicians, Nina Bary and D. Menchoff, in the early 20th century completely solved the problem of precisely what functions can be expressed as the composition of two absolutely continuous functions.

My main motivation for answering this simple question is to encourage you to read Saks' excellent account of this interesting research. I would guess most analysis students may never have heard of it. Most know about Lusin's condition (N), but probably not about Banach's conditions (T${}_1$), (T${}_2$), and (S).

Who are Nina Bary and D. Menchoff? Here is a photo of the Moscow State University mathematicians from the 1950s. The marvelous Nina sits between a rather sour looking Menshov and an overly cheerful Tolstov. Tolstov, in spite of looking like a KGB colonel ordering a group photo, was a good mathematician himself.