Proof of The Earlier and the New Versions of the Definitions of Probability Spaces, Laws, Expectation, Independence and Modes of Convergence 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. Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, Probability Spaces: Notation and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, 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 Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, Probability Spaces: Notation, Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, 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); and the real matrices, which the new settings take from Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i), by The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (ii). In addition, Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity reads real numbers as in The Real Numbers as a Type where the earlier version follows The Real Numbers as a Type, and reads the reciprocal of a nonzero element of $\mathrm{Real}$ by The Real Numbers as a Type (vii) where the earlier version cites The Reciprocal on the Type of Real Numbers (i); both readings take a real number, and its reciprocal, to the element with the same set, so they 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 (v).

(i): measurable spaces, measures, Borel $\sigma$-algebras and measurable maps, by The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (i), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (iii), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (ii) and The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (iv); metric spaces, by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (i); and $\mathbb{R}^{n}$, by The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (i). By the first part, $(\Omega,\mathcal{F},P)$ is a probability space in the new version too, so its events and random elements are defined in both.

(ii): probability spaces and random elements, by (i), the random element $Y$ with which $X$ is compared included ((i), applied to $(\Omega',\mathcal{F}',P')$ in place of $(\Omega,\mathcal{F},P)$); Borel $\sigma$-algebras, by The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (ii); metric spaces, by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (i).

(iii): probability spaces, random variables and random vectors, by (i); $\mathbb{R}^{n}$ 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); $\mathcal{M}^{+}$, $\mathcal{L}^{1}$ and their integrals come from the same setting Measurable Functions and the Lebesgue Integral: Standing Notation in both versions.

(iv): metric spaces, by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (i); events and random elements, by (i); identical distribution, by (ii); sequences over $\mathbb{N}$, by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i); and the $\sigma$-algebras $\sigma(X)$ (σ-Algebras Generated by Random Elements (i)) and the independence of classes of events and of $\sigma$-algebras (Independence of Classes of Events and of σ-Algebras (i), Independence of Classes of Events and of σ-Algebras (ii), Independence of Classes of Events and of σ-Algebras (iii)) are the same published items in both versions.

(v): separable metric spaces, by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (ii) and The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (i); sequences and their convergence, by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (ii); the random elements and probabilities of the setting, by (i).

(vi): metric spaces and convergence in them, 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 (iii); sequences, by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i); the random elements and events of the setting, by (i).

(vii): sequences and their convergence, by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (ii); $\mathbb{R}^{k}$, by The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (i); the random vectors and expectations of the setting, by (i) and (iii).

(viii): metric spaces, by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (i); Borel $\sigma$-algebras, by The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (ii); sequences and their convergence, by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (ii); finite measures, by The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (iii); and $C_{b}(S)$ (Bounded Continuous Functions on a Metric Space (i)) and the integrals against it, from Measurable Functions and the Lebesgue Integral: Standing Notation, are the same published items in both versions, the continuity they involve agreeing by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (vii).

(ix): metric spaces, by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (i); weak convergence, by (viii); the random elements and laws of the setting, by (i) and (ii), and likewise the random elements $X_{n}$ compared with $X$ and their laws ((i) and (ii), applied to each $(\Omega_{n},\mathcal{F}_{n},P_{n})$ in place of $(\Omega,\mathcal{F},P)$).