The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree

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

(i) $\sigma$-algebras and measurable spaces. For all sets $X$ and $\mathcal{F}$, $\mathcal{F}$ is a $\sigma$-algebra on $X$ in the earlier version if and only if it is a $\sigma$-algebra on $X$ in the new version; so $(X,\mathcal{F})$ is a measurable space in the earlier version if and only if it is a measurable space in the new one, with the same measurable sets.

(ii) Borel $\sigma$-algebras. For every metric space $(X,d)$, its Borel $\sigma$-algebra in the earlier version is its Borel $\sigma$-algebra in the new version; and for all $n,p,q\in\mathbb{N}$ the two versions give the same $\mathcal{B}(\mathbb{R})$, $\mathcal{B}(\mathbb{R}^{n})$ and $\mathcal{B}(\mathbb{R}^{p\times q})$ (Borel Sigma-Algebra of a Metric Space (ii), Borel Sigma-Algebra of a Metric Space (iii), Borel Sigma-Algebra of a Metric Space (iv); Borel Sigma-Algebra of a Metric Space (ii), Borel Sigma-Algebra of a Metric Space (iii), Borel Sigma-Algebra of a Metric Space (iv)).

(iii) Measures. For every measurable space $(X,\mathcal{F})$ and every map $\mu:\mathcal{F}\to[0,\infty]$, $\mu$ is a measure on $(X,\mathcal{F})$ in the earlier version if and only if it is a measure in the new version; and then $(X,\mathcal{F},\mu)$ is a measure space, and $\mu$ is finite, $\sigma$-finite or a probability measure, in the earlier version if and only if it is a measure space, finite, $\sigma$-finite or a probability measure in the new one.

(iv) Measurable maps. For all measurable spaces $(X,\mathcal{F})$ and $(Y,\mathcal{G})$, a map $f:X\to Y$ is $\mathcal{F}/\mathcal{G}$-measurable in the earlier version if and only if it is $\mathcal{F}/\mathcal{G}$-measurable in the new version, so the two versions give the same set $\mathrm{Meas}(X,\mathcal{F};Y,\mathcal{G})$ (Measurable Map, and the Set of Measurable Maps (ii), Measurable Map, and the Set of Measurable Maps (ii)). Likewise, for every measurable space $(X,\mathcal{F})$, all $n,p,q\in\mathbb{N}$ and each of the sets $Z=\mathbb{R}$, $\mathbb{R}^{n}$, $\mathbb{R}^{p\times q}$ and $[0,\infty]$, a map $X\to Z$ is measurable in the earlier version (Measurable Map, and the Set of Measurable Maps (iii), Measurable Map, and the Set of Measurable Maps (iv), Measurable Map, and the Set of Measurable Maps (v), Measurable Map, and the Set of Measurable Maps (vi)) if and only if it is measurable in the new one (Measurable Map, and the Set of Measurable Maps (iii), Measurable Map, and the Set of Measurable Maps (iv), Measurable Map, and the Set of Measurable Maps (v), Measurable Map, and the Set of Measurable Maps (vi)).

(v) Integrals of nonnegative maps. For every measurable space $(X,\mathcal{F})$ and every measure $\mu$ on it, the set $\mathcal{M}^{+}(X,\mathcal{F})$ of the earlier version is the set $\mathcal{M}^{+}(X,\mathcal{F})$ of the new version, and for every $f$ in it the integral $\int_{X}f\,d\mu$ in the earlier version is its integral in the new one.

(vi) Integrable functions. For every measure space $(X,\mathcal{F},\mu)$, a measurable function $f:X\to\mathbb{R}$ is integrable with respect to $\mu$ in the earlier version if and only if it is integrable in the new version; so the two versions give the same set $\mathcal{L}^{1}(\mu)$ (Integrable Functions and Their Lebesgue Integral (ii)) and, for every $f$ in it, the same integral $\int_{X}f\,d\mu$ (Integrable Functions and Their Lebesgue Integral (iii)).