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

Each new version treated here has the text of its earlier version, except that some references point to newer versions, which agree with the earlier ones by an earlier clause of this item or by the bridge helpers cited below; a moved reference to a result states a fact and fixes no notion. So each clause holds once the notions its definitions cite are known to agree.

The moved references include the settings. Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation and Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, and the newer versions Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces and The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers that they extend, differ from Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces and The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers in this way only. Among the notions they fix, the natural and complex numbers agree by The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (i) and The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (ii); finite sums and products, which the new settings take from Iterated Operations: Finite Sums and Finite Products (iii), by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (vi); $\mathbb{R}^{n}$ and the real matrices, by The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (i) and The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (ii); and $\sigma$-algebras, Borel $\sigma$-algebras and measures, which Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation (iv), Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation (v) and Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation (vi) take from the new versions treated here, by (i), (ii) and (iii).

(i): sequences, by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i).

(ii): metric spaces and their open sets, by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (i) and The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (v).

(iii): measurable spaces and sequences, by (i) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i).

(iv): measurable spaces and Borel $\sigma$-algebras, by (i) and (ii); the notion of measure that the adopted settings fix (Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation (vi), Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation (vi)), by (iii); and $\mathbb{R}^{n}$ and the real matrices, which the new settings take from the new versions of their definitions, by The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (i) and The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (ii).

(v): measurable spaces, measures and measurable maps, by (i), (iii) and (iv). Both versions take the set $\mathcal{S}^{+}$ of nonnegative simple functions and the simple integral $I_{\mu}$ from the same item, Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iii) and Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iv), stated over the earlier notions of measurable space and measure, which agree with the new ones by (i) and (iii). So the set $\Sigma_{f}$ of simple integrals below $f$, and its supremum $\int_{X}f\,d\mu$, are the same in both versions.

(vi): measure spaces, measurable functions and integrals of nonnegative maps, by (iii), (iv) and (v).