Measurable-Space Bookkeeping: Borel σ-Algebras of Metric Spaces, Closed Subintervals in the Restricted Borel σ-Algebra, and the Whole Space Almost Everywhere

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation:

(i) Let $(S,d)$ be a metric space. Then its Borel $\sigma$-algebra $\mathcal{B}(S)$ is a $\sigma$-algebra on $S$.

(ii) Let $a,b,s,t\in\mathbb{R}$ with $a\le s$, $s\le t$ and $t\le b$, and let

$$\mathcal{B}(\mathbb{R})_{[a,b]}=\{B\in\mathcal{B}(\mathbb{R}):B\subseteq[a,b]\}$$

be the restriction of $\mathcal{B}(\mathbb{R})$ to the closed interval $[a,b]$. Then $[s,t]\in\mathcal{B}(\mathbb{R})_{[a,b]}$.

(iii) Let $(X,\mathcal{F},\mu)$ be a measure space. Then $X$ holds $\mu$-almost everywhere.