A step in the proof of $\{N\text{-bounded variation}\}\subseteq\{\text{bounded variation}\}$

Question

I have a question about the proof of Lemma 1 in this paper by Josephy, which discusses necessary and sufficient conditions on $g$ such that $f\circ g$ (respectively $g\circ f$) is a function of bounded variation for all $f$ of bounded variation. I have copied the statement of the lemma and a part of the proof here for convenience.

Definition: A function $f : [0, 1]\rightarrow\mathbb{R}$ is said to be of $N$-bounded variation if $f^{-1}([a, b])$ can be written as a union of $N$ or fewer intervals (where we allow singletons as degenerate closed intervals) for all $[a, b]\subseteq\mathbb{R}$.

Lemma 1: Let $h$ be a function of $N$-bounded variation such that $|h(x)|\leq M$ for all $x\in[0, 1]$. Then $\text{Var}\:h\leq 4M(N+1)$.

Proof: If $\text{Var}\:h > 4M(N+1)$, then there exists a partition $\{x_{0},\dots, x_{n}\}$ of $[0, 1]$ with $$\sum_{i = 1}^{n}|h(x_{i})-h(x_{i-1})| > 4M(N+1).$$ Let $J_{i}$ denote the closed interval with endpoints $h(x_{i-1})$ and $h(x_{i})$. Some $[a', b']\subseteq [-M, M]$ with $a' < b'$ is covered more than $2(N+1)$ times by intervals of the form $J_{i}$.

I am having trouble proving the last statement above. Proving this amounts to showing that there exist indices $i_{1} < \cdots < i_{\ell}$ with $\ell > 2(N+1)$ such that $\bigcap_{k = 1}^{\ell}J_{i_{k}}$ is a non-degenerate closed interval. (Note that the displayed equation above implies that $n > 2(N+1)$.)

My thoughts:

I have tried using the inclusion-exclusion principle, but it does not seem to help. One thing to be noted is that consecutive intervals $J_{i}$ and $J_{i+1}$ share at least one endpoint. With this in mind, I have tried examining the level sets of the piecewise-linear function $f : [0, 1]\rightarrow \mathbb{R}$ with $f(x_{i}) = h(x_{i})$ for each $i$. Intuitively, since $\sum_{i =1}^{n}|J_{i}|$ is large, if the function is initially monotonically increasing (say), then it must eventually decrease, and so on. (Another way of thinking about this is that we're trying to pack a long string of length $\sum_{i = 1}^{n} |J_{i}|$ in a "box" of length $2M$ by folding it. We need the string to overlap sufficiently many times.) However, I have not been able to translate these ideas into a rigorous proof.

Edit: Here's another observation that might help: we may assume without loss of generality that $h(x_{0}) < h(x_{1})$, $h(x_{1}) > h(x_{2})$, $h(x_{2}) < h(x_{3})$, and so on.

Any help is appreciated. Thank you.

Answer 1

Suppose that
$$\displaystyle 4M(N+1) < \sum_{i = 1}^{n}|h(x_{i})-h(x_{i-1})| = \int_{-M}^M \sum_{i = 1}^{n} 1_{J_i}(t) \ dt,$$ where either $J_i=[h(x_{i-1}),h(x_i)]$ if $h(x_{i-1})\leq h(x_i)$ or $J_i=[h(x_{i}),h(x_{i-1})]$ otherwise, and $$1_{J_i}(t):= \begin{cases} 1, & \text{if } t \in J_i,\\ 0, & \text{if } t \not\in J_i. \end{cases}$$

If almost every point of $[-M,M]$ is covered at most $2(N+1)$ times, it follows that
$$u(t)=\sum_{i = 1}^{n} 1_{J_i}(t)\leq 2(N+1)$$
for almost every $t\in [-M,M]$, so
$$\int_{-M}^M \sum_{i = 1}^{n} 1_{J_i}(t) \ dt\leq 2M \ \left( ess \ sup_{t\in [-M,M]} u(t)\right) \leq 4M(N+1).$$

Here $ess \ sup$ is the essential supremum (https://en.wikipedia.org/wiki/Essential_infimum_and_essential_supremum) on $[-M,M]$. So there is a set of positive Lebesgue measure (positive length) where $u(t) > 2(N+1).$

But the function $u$ can be written as (see https://en.wikipedia.org/wiki/Step_function)
$$u(t)=\sum_{k=0}^m c_k 1_{A_k}$$
for disjoint intervals $A_k$ and natural numbers $c_k$, so one of these intervals $A_{k_0}$ is non-degenerate and $c_{k_0} > 2(N+1)$ there.