Proof of The Borel Sets of ℝ Form a σ-Algebra, and Integrable Functions Are Measurable Real-Valued Maps

Proof of (i). By Borel Sigma-Algebra of a Metric Space (ii) and Borel Sigma-Algebra of a Metric Space (i),

$$\mathcal{B}(\mathbb{R})=\sigma(\mathcal{T}_{d_{\mathbb{R}}}),$$

the $\sigma$-algebra generated by the set $\mathcal{T}_{d_{\mathbb{R}}}$ of open subsets of $\mathbb{R}$, which is a set of subsets of $\mathbb{R}$. By Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), taking the set to be $\mathbb{R}$, it is a $\sigma$-algebra on $\mathbb{R}$, so $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is a measurable space.

Proof of (ii) and (iii). By Integrable Functions and Their Lebesgue Integral (ii),

$$f\in\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$$

(formed for the measurable space $(X,\mathcal{F})$ of Measure, Measure Space, and Probability Measure (ii) and that of (i)). By Measurable Map, and the Set of Measurable Maps (ii) its elements are exactly the $\mathcal{F}/\mathcal{B}(\mathbb{R})$-measurable maps $X\to\mathbb{R}$, which gives (ii) and, by Measurable Map, and the Set of Measurable Maps (iii), (iii). $\blacksquare$