The Nondecreasing Envelope of a Truncated Modulus of Continuity

In the setting of The Real Numbers: Standing Notation and Background, let $T=\{t\in\mathbb{R}:0\le t\}$, let $\omega:T\to\mathbb{R}$ be a modulus of continuity, let $M\in\mathbb{R}$ satisfy $0\le M$, and let $\min\{a,b\}$ denote the minimum of two real numbers. For $s\in T$ let

this set is nonempty, since it contains $\min\{\omega(0),M\}$, and is bounded above by $M$, by claim 1 of Elementary Properties of the Minimum of Two Elements, so its supremum $\sup\Omega(s)$ is defined. ¶ Let $\bar{\omega}:T\to\mathbb{R}$ be the function with value $\bar{\omega}(s)=\sup\Omega(s)$ at $s\in T$, called the nondecreasing envelope of $\omega$ truncated at $M$. Then the following hold.

1. (Modulus of continuity)¶ $\bar{\omega}$ is a modulus of continuity, and $0\le\bar{\omega}(s)\le M$ for every $s\in T$.

2. (Monotonicity)¶ $\bar{\omega}(s)\le\bar{\omega}(s')$ for all $s,s'\in T$ with $s\le s'$.

3. (Truncated majorant)¶ $\min\{\omega(t),M\}\le\bar{\omega}(t)$ for every $t\in T$. In particular, if $a\in\mathbb{R}$ satisfies $a\le\omega(t)$ and $a\le M$, then $a\le\bar{\omega}(t)$.