Proof of Joint Distribution, Expectations, and Block Independence for Independent Random Variables

We keep the notation of the statement. Throughout, for $1\le k\le r$ let $\mathcal{P}_k$ denote the family of iterated rectangles $B_1\times\cdots\times B_k\subseteq\mathbb{R}^k$ with all $B_i\in\mathcal{B}(\mathbb{R})$ (Cartesian products, under the identification fixed in the statement). $\mathcal{P}_k$ is a $\pi$-system in the sense of Dynkin's Pi-Lambda Theorem — it is closed under intersection, since $(B_1\times\cdots\times B_k)\cap(B'_1\times\cdots\times B'_k)=(B_1\cap B'_1)\times\cdots\times(B_k\cap B'_k)$ — and it contains $\mathbb{R}^k$.

Step 0 (iterated rectangles generate, and product values). We show by induction on $k$ that the generated $\sigma$-algebra satisfies $\sigma(\mathcal{P}_k)=\mathcal{B}_k$. For $k=1$, $\mathcal{P}_1=\mathcal{B}(\mathbb{R})$. Let $k\ge2$ and assume $\sigma(\mathcal{P}_{k-1})=\mathcal{B}_{k-1}$. Each member of $\mathcal{P}_k$ is a measurable rectangle in the sense of Product Sigma-Algebra (its first factor lies in $\mathcal{P}_{k-1}\subseteq\mathcal{B}_{k-1}$), so $\sigma(\mathcal{P}_k)\subseteq\mathcal{B}_k$. Conversely, fix $B\in\mathcal{B}(\mathbb{R})$ and let $\mathcal{A}_B=\{A\subseteq\mathbb{R}^{k-1}:A\times B\in\sigma(\mathcal{P}_k)\}$. Then $\mathcal{A}_B$ is a $\sigma$-algebra: it contains $\mathbb{R}^{k-1}$, is closed under countable unions ($(\bigcup_j A_j)\times B=\bigcup_j(A_j\times B)$), and under complements, since $(\mathbb{R}^{k-1}\setminus A)\times B=(\mathbb{R}^{k-1}\times B)\setminus(A\times B)$ with $\mathbb{R}^{k-1}\times B\in\mathcal{P}_k$. It contains $\mathcal{P}_{k-1}$, hence contains $\sigma(\mathcal{P}_{k-1})=\mathcal{B}_{k-1}$. So every measurable rectangle $A\times B$ lies in $\sigma(\mathcal{P}_k)$, and these generate $\mathcal{B}_k$; thus $\mathcal{B}_k\subseteq\sigma(\mathcal{P}_k)$.

Also, by induction on $k$ using the defining property of the product measure in Existence and Uniqueness of the Product Measure,

$$(\nu_1\otimes\cdots\otimes\nu_k)(B_1\times\cdots\times B_k)=\prod_{i=1}^{k}\nu_i(B_i),$$

with the finite product notation.

Step 1 (Claim 1). Measurability of $V$: by induction on $k$, with $V^{(k)}=(V_1,\dots,V_k)$. The class $\{C\subseteq\mathbb{R}^k:(V^{(k)})^{-1}(C)\in\mathcal{F}\}$ is a $\sigma$-algebra (preimages commute with complements and countable unions) containing every measurable rectangle $A\times B$, since $(V^{(k)})^{-1}(A\times B)=(V^{(k-1)})^{-1}(A)\cap V_k^{-1}(B)\in\mathcal{F}$ by the induction hypothesis and measurability of $V_k$; hence it contains $\mathcal{B}_k$.

$P_V$ is a probability measure on $(\mathbb{R}^r,\mathcal{B}_r)$: $P_V(\emptyset)=0$, $P_V(\mathbb{R}^r)=P(\Omega)=1$, and countable additivity follows from that of $P$ because preimages of pairwise disjoint sets are pairwise disjoint and $V^{-1}(\bigcup_j C_j)=\bigcup_j V^{-1}(C_j)$.

On $\mathcal{P}_r$: by independence of $V_1,\dots,V_r$ and Step 0,

$$P_V(B_1\times\cdots\times B_r)=P\Bigl(\bigcap_{i=1}^{r}\{V_i\in B_i\}\Bigr)=\prod_{i=1}^{r}P(V_i\in B_i)=\prod_{i=1}^{r}\nu_i(B_i)=(\nu_1\otimes\cdots\otimes\nu_r)(B_1\times\cdots\times B_r).$$

Let $\mathcal{L}=\{C\in\mathcal{B}_r:P_V(C)=(\nu_1\otimes\cdots\otimes\nu_r)(C)\}$. Since both set functions are probability measures, $\mathcal{L}$ contains $\mathbb{R}^r$, is closed under proper differences (if $A\subseteq B$ both lie in $\mathcal{L}$, then $B\setminus A\in\mathcal{L}$ by additivity and finiteness) and under increasing countable unions (continuity from below, obtained by writing an increasing union as a disjoint union of successive differences and using countable additivity). So $\mathcal{L}$ is a $\lambda$-system containing the $\pi$-system $\mathcal{P}_r$, and Dynkin's Pi-Lambda Theorem together with Step 0 gives $\mathcal{L}\supseteq\sigma(\mathcal{P}_r)=\mathcal{B}_r$. This proves Claim 1.

Step 2 (Claim 2). $\varphi\circ V$ is a random variable since preimages compose: $(\varphi\circ V)^{-1}(B)=V^{-1}(\varphi^{-1}(B))$. The identity $\mathbb{E}[\varphi\circ V]=\int_{\mathbb{R}^r}\varphi\,dP_V$ follows by the standard machine exactly as in the proof of Change of Variables for Expectations, with $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ there replaced by $(\mathbb{R}^r,\mathcal{B}_r)$: for indicators both sides equal $P_V(C)$ by the integral of a simple function; linearity for nonnegative functions (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) extends this to nonnegative simple $\varphi$; for nonnegative measurable $\varphi$, compose with the dyadic staircase functions of Step 0(b) of the proof of Linearity and Monotonicity of the Lebesgue Integral and apply Monotone Convergence Theorem on both spaces. If instead $\varphi$ is bounded, say $|\varphi|\le M$, apply the nonnegative case to the positive and negative parts $\varphi^{\pm}$ (as in Integrable Function and the Lebesgue Integral); each is bounded by $M$, whose integral against a probability measure is $M<\infty$, so $\varphi\circ V$ is integrable and the identities subtract. By Claim 1, $P_V=\nu_1\otimes\cdots\otimes\nu_r$, so the value depends only on $\varphi$ and $\nu_1,\dots,\nu_r$.

Step 3 (Claim 3). Selecting a subfamily of an independent family preserves independence (immediate from Independence of Events and of Random Variables), so the entries of the tuples $V_I=(V_{i_1},\dots,V_{i_p})$ and $V_J=(V_{j_1},\dots,V_{j_q})$, and of the combined tuple over $I\cup J$, are independent, and each tuple map is measurable by Step 1 (whose measurability argument uses no independence). Hence $\varphi(V_I)$ and $\psi(V_J)$ are random variables by Step 2. Fix Borel sets $B,B'\subseteq\mathbb{R}$ and put $C=\varphi^{-1}(B)\in\mathcal{B}_p$, $D=\psi^{-1}(B')\in\mathcal{B}_q$. It suffices to prove

$$P(V_I\in C,\ V_J\in D)=P(V_I\in C)\,P(V_J\in D)\qquad\text{for all }C\in\mathcal{B}_p,\ D\in\mathcal{B}_q;$$

independence of the two random variables $\varphi(V_I)$, $\psi(V_J)$ in the sense of Independence of Events and of Random Variables then follows (for a pair, only the product identity for the pair itself is nontrivial).

If $C$ and $D$ are iterated rectangles, both sides expand by independence of the combined family into the same product of the numbers $P(V_i\in\cdot)$ over $I\cup J$. Now fix an iterated rectangle $D$. The two set functions $C\mapsto P(V_I\in C,\ V_J\in D)$ and $C\mapsto P(V_I\in C)P(V_J\in D)$ are finite measures on $\mathcal{B}_p$ (countable additivity as in Step 1) with the same total mass $P(V_J\in D)$, agreeing on $\mathcal{P}_p$; the class where they agree is a $\lambda$-system exactly as in Step 1, so by Dynkin's Pi-Lambda Theorem and Step 0 they agree on all of $\mathcal{B}_p$. Finally fix an arbitrary $C\in\mathcal{B}_p$ and repeat the same argument in the $D$-coordinate over the $\pi$-system $\mathcal{P}_q$. This proves Claim 3.

Step 4 (Claim 4). Projections: $\pi_r^{-1}(B)=\mathbb{R}^{r-1}\times B\in\mathcal{B}_r$, and for $i<r$, by induction on $r$, $\pi_i^{-1}(B)=A\times\mathbb{R}$ with $A\in\mathcal{B}_{r-1}$, a measurable rectangle. Next, if $u,v:\mathbb{R}^r\to\mathbb{R}$ are jointly Borel then so is $u+v$: for $t\in\mathbb{R}$, by density of the rationals,

$$\{x:u(x)+v(x)>t\}=\bigcup_{s\in\mathbb{Q}}\bigl(\{x:u(x)>s\}\cap\{x:v(x)>t-s\}\bigr),$$

a countable union of members of $\mathcal{B}_r$; and the rays $(t,\infty)$ generate $\mathcal{B}(\mathbb{R})$ (every open interval with rational endpoints is obtained from rays by countable set operations, e.g. $(c,d)=(c,\infty)\setminus\bigcap_{k\ge1}(d-1/k,\infty)$, and every open set is a countable union of such intervals by density of the rationals), so the generator criterion of Measurable Function and Real-Valued Measurable Function applies. The addition map is the sum of the $r$ projections, hence jointly Borel by induction on the number of summands. Finally, $(t\circ\varphi)^{-1}(B)=\varphi^{-1}(t^{-1}(B))$ with $t^{-1}(B)\in\mathcal{B}(\mathbb{R})$, proving the composition statement. $\blacksquare$