Proof of Independent Random Elements: Preimages, Functions and Blocks

Throughout, for a random element $Z$ of a metric space $R$ and $B\in\mathcal{B}(R)$, $\{Z\in B\}=Z^{-1}(B)$, and for random elements $Z_{1},\dots,Z_{n}$ the event $\{Z_{1}\in B_{1},\dots,Z_{n}\in B_{n}\}$ is $\bigcap_{k=1}^{n}Z_{k}^{-1}(B_{k})$ (Probability Spaces: Notation (iii)); and a family indexed by $[2]=\{1,2\}$ is given by its two terms (Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i), Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (ii)).

Preimage classes. For a random element $Z$ of a metric space $R$ let $\mathcal{C}_{Z}=\{Z^{-1}(B):B\in\mathcal{B}(R)\}$, so that $\sigma(Z)=\sigma(\mathcal{C}_{Z})$ and $\mathcal{C}_{Z}\subseteq\sigma(Z)\subseteq\mathcal{F}$ (σ-Algebras Generated by Random Elements (i), Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), The σ-Algebra Generated by Measurable Functions into Metric Spaces (i)). Then $\mathcal{C}_{Z}$ is a $\pi$-system on $\Omega$: for $B,B'\in\mathcal{B}(R)$,

$$Z^{-1}(B)\cap Z^{-1}(B')=Z^{-1}(B\cap B'),$$

and $B\cap B'\in\mathcal{B}(R)$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix)).

Proof of (i). For $j\in J$ write $\mathcal{C}_{j}=\mathcal{C}_{X_{j}}$. If $(X_{j})_{j\in J}$ is independent, that is, $(\sigma(X_{j}))_{j\in J}$ is independent (Independence of Events and of Random Elements (ii)), the displayed condition is its product formula for the events $X_{j_{k}}^{-1}(B_{k})\in\sigma(X_{j_{k}})$.

Conversely, assume the condition; we show that $(\mathcal{C}_{j})_{j\in J}$ is independent. Let $n\in\mathbb{N}$, let $j_{1},\dots,j_{n}\in J$ be distinct and let $A_{k}\in\mathcal{C}_{j_{k}}$ for $k\in[n]$. We show by induction on $m\in\{0,\dots,n\}$ (Induction for Predicative Formulas, and the Order of ℕ below 1 and below n+1 (i)): for all $B_{m+1}\in\mathcal{B}(S_{j_{m+1}}),\dots,B_{n}\in\mathcal{B}(S_{j_{n}})$, the product formula holds for the events $A_{1},\dots,A_{m},X_{j_{m+1}}^{-1}(B_{m+1}),\dots,X_{j_{n}}^{-1}(B_{n})$. For $m=0$ this is the condition. If it holds for some $m<n$, then $A_{m+1}=X_{j_{m+1}}^{-1}(B)$ for some $B\in\mathcal{B}(S_{j_{m+1}})$, and the case $m$ with $B_{m+1}=B$ gives the case $m+1$. The case $m=n$ is the product formula for $A_{1},\dots,A_{n}$. Hence $(\mathcal{C}_{j})_{j\in J}$ is an independent family of $\pi$-systems contained in $\mathcal{F}$, and $(\sigma(\mathcal{C}_{j}))_{j\in J}=(\sigma(X_{j}))_{j\in J}$ is independent by Independence of σ-Algebras: π-Systems, Grouping and Pairs (ii), that is, $(X_{j})_{j\in J}$ is independent.

Proof of (ii). If $X$ and $Y$ are independent, that is, $\sigma(X)$ and $\sigma(Y)$ are independent (Independence of Events and of Random Elements (iii)), the displayed identity is the product rule for the events $X^{-1}(B)\in\sigma(X)$ and $Y^{-1}(B')\in\sigma(Y)$. Conversely, the identity says that the $\pi$-systems $\mathcal{C}_{X}$ and $\mathcal{C}_{Y}$ satisfy $P(C\cap D)=P(C)P(D)$ for $C\in\mathcal{C}_{X}$ and $D\in\mathcal{C}_{Y}$, so $\sigma(\mathcal{C}_{X})=\sigma(X)$ and $\sigma(\mathcal{C}_{Y})=\sigma(Y)$ are independent by Independence of σ-Algebras: π-Systems, Grouping and Pairs (v).

Proof of (iii). The families $(\sigma(X_{j}))_{j\in J}$, $(g_{j}(X_{j}))_{j\in J}$ and $(\sigma(g_{j}(X_{j})))_{j\in J}$ exist (Borel Sets of a Variable Metric Space as a Predicative Condition, and the Families of σ-Algebras Generated by Random Elements and of Their Composites (vi), Borel Sets of a Variable Metric Space as a Predicative Condition, and the Families of σ-Algebras Generated by Random Elements and of Their Composites (vii)), the last since each $g_{j}(X_{j})$ is a random element of $T_{j}$ (Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii)). Let $j\in J$. $X_{j}$ is $\sigma(X_{j})/\mathcal{B}(S_{j})$-measurable (The σ-Algebra Generated by Measurable Functions into Metric Spaces (i)), so $g_{j}(X_{j})=g_{j}\circ X_{j}$ is $\sigma(X_{j})/\mathcal{B}(T_{j})$-measurable (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (ii)), and therefore $\sigma(g_{j}(X_{j}))\subseteq\sigma(X_{j})$ by the minimality in The σ-Algebra Generated by Measurable Functions into Metric Spaces (i). As $(\sigma(X_{j}))_{j\in J}$ is independent, so is $(\sigma(g_{j}(X_{j})))_{j\in J}$ by Independence of σ-Algebras: π-Systems, Grouping and Pairs (i), that is, $(g_{j}(X_{j}))_{j\in J}$ is independent.

Proof of (iv). As $\sigma(X)$ and $\sigma(Y)$ are independent, the family of $\sigma$-algebras $(\sigma(X),\sigma(Y))$ indexed by $[2]$ is independent by Independence of σ-Algebras: π-Systems, Grouping and Pairs (iv), that is, the family $(X,Y)$ of random elements of $S$ and $T$ indexed by $[2]$ is independent. By (iii), applied to $g$ and $h$, the family $(g(X),h(Y))$ indexed by $[2]$ is independent, that is, so is the family $(\sigma(g(X)),\sigma(h(Y)))$; by the equivalence (Independence of σ-Algebras: π-Systems, Grouping and Pairs (iv)), $\sigma(g(X))$ and $\sigma(h(Y))$ are independent, so $g(X)$ and $h(Y)$ are independent (Independence of Events and of Random Elements (iii)).

Proof of (v). The family $(\sigma(X_{k}))_{k\in[n]}$ is independent. Apply Independence of σ-Algebras: π-Systems, Grouping and Pairs (iii) with $K=[2]$, $J_{1}=[m]$ and $J_{2}=\{m+1,\dots,n\}$ (Intervals of Natural Numbers (i)), which are disjoint: with $\mathcal{H}_{1}=\sigma\bigl(\bigcup_{k\in[m]}\sigma(X_{k})\bigr)$ and $\mathcal{H}_{2}=\sigma\bigl(\bigcup_{k=m+1}^{n}\sigma(X_{k})\bigr)$, the family $(\mathcal{H}_{1},\mathcal{H}_{2})$ is independent, so $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are independent by the equivalence (Independence of σ-Algebras: π-Systems, Grouping and Pairs (iv)). The components of $U$ are $U_{k}=X_{k}$ for $k\in[m]$ (Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (iii)). Each such $X_{k}$ is $\sigma(X_{k})/\mathcal{B}(\mathbb{R})$-measurable (The σ-Algebra Generated by Measurable Functions into Metric Spaces (i)) and $\sigma(X_{k})\subseteq\mathcal{H}_{1}$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii)), so it is $\mathcal{H}_{1}/\mathcal{B}(\mathbb{R})$-measurable. By Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i), applied on the measurable space $(\Omega,\mathcal{H}_{1})$, $U$ is $\mathcal{H}_{1}/\mathcal{B}(\mathbb{R}^{m})$-measurable, and so $\sigma(U)\subseteq\mathcal{H}_{1}$ by The σ-Algebra Generated by Measurable Functions into Metric Spaces (i). In the same way, as $V_{l}=X_{m+l}$ for $l\in[n-m]$, $\sigma(V)\subseteq\mathcal{H}_{2}$. Hence $P(A\cap B)=P(A)P(B)$ for $A\in\sigma(U)$ and $B\in\sigma(V)$, that is, $U$ and $V$ are independent (Independence of Classes of Events and of σ-Algebras (iii), Independence of Events and of Random Elements (iii)).