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

Proof of (i). By Borel Sigma-Algebra of a Metric Space (i), $\mathcal{B}(S)=\sigma(\mathcal{T}_{d})$, the $\sigma$-algebra generated by the set $\mathcal{T}_{d}$ of open subsets of $S$, and this is a $\sigma$-algebra on $S$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii).

Proof of (ii). By Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii), $[s,t]\in\mathcal{B}(\mathbb{R})$. Every $r\in[s,t]$ satisfies $s\le r\le t$ (Intervals of the Real Line (ii)), hence

$$a\le s\le r\le t\le b$$

and $r\in[a,b]$. So $[s,t]\subseteq[a,b]$, that is, $[s,t]\in\mathcal{B}(\mathbb{R})_{[a,b]}$.

Proof of (iii). By A Property Holding Almost Everywhere (i) we must show that $X\setminus X$ is $\mu$-null. Now $X\setminus X=\emptyset$, and $\emptyset\in\mathcal{F}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (vi) ($\mathcal{F}$ is a $\sigma$-algebra on $X$ by Measure, Measure Space, and Probability Measure (ii)) with $\mu(\emptyset)=0$ by Measure, Measure Space, and Probability Measure (i), so $X\setminus X$ is $\mu$-null by Null Set of a Measure (i), taking $B=\emptyset$. $\blacksquare$