A Function Nondecreasing on a Borel Subset of the Real Line and Vanishing Outside It is Borel

In the setting of The Real Numbers: Standing Notation and Background, let $\mathcal{B}(\mathbb{R})$ be the Borel $\sigma$-algebra of the real line, and let measurable mean measurable with respect to $\mathcal{B}(\mathbb{R})$ and $\mathcal{B}(\mathbb{R})$.

Let $E\in\mathcal{B}(\mathbb{R})$ and let $T:\mathbb{R}\to\mathbb{R}$.

1. (Borel measurability)¶ Suppose that $T(x)\le T(x')$ for all $x,x'\in E$ with $x\le x'$, and that $T(x)=0$ for every $x\in\mathbb{R}\setminus E$. Then $T$ is measurable.