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

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

(i) $\mathcal{B}(\mathbb{R})$ is a $\sigma$-algebra on $\mathbb{R}$, so $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is a measurable space.

Let $(X,\mathcal{F},\mu)$ be a measure space, let $\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$ be the set of measurable maps of Measurable Map, and the Set of Measurable Maps (ii), let $\mathcal{L}^{1}(X,\mathcal{F},\mu)$ be the set of integrable functions, and let $f\in\mathcal{L}^{1}(X,\mathcal{F},\mu)$.

(ii) $f$ is a map $f:X\to\mathbb{R}$.

(iii) $f\in\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$, and $f$ is measurable as a map into $\mathbb{R}$.