Throughout, $r$ is a \reftext{def:natural-numbers-2026a}{natural number} with $r\ge 1$, $\mathbb{R}^r$ is \reftext{def:euclidean-space-rn-2026a}{Euclidean space}, $\mathbb{R}$ denotes the \reftext{def:real-numbers-c54-2026c}{real numbers}, and $\mathcal{B}(\mathbb{R})$ the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra}.

Define the \textbf{$r$-fold product Borel $\sigma$-algebra} $\mathcal{B}_r$ on $\mathbb{R}^r$ iteratively: $\mathcal{B}_1=\mathcal{B}(\mathbb{R})$ and, for $2\le k\le r$, $\mathcal{B}_k=\mathcal{B}_{k-1}\otimes\mathcal{B}(\mathbb{R})$, the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$-algebra} on $\mathbb{R}^{k}$, identifying the \reftext{def:cartesian-product-sets-2026a}{Cartesian product} $\mathbb{R}^{k-1}\times\mathbb{R}$ with $\mathbb{R}^{k}$ via $((x_1,\dots,x_{k-1}),x_k)\mapsto(x_1,\dots,x_k)$. Given probability \reftext{def:measure-measure-space-2026a}{measures} $\nu_1,\dots,\nu_r$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, define the probability measure $\nu_1\otimes\cdots\otimes\nu_r$ on $(\mathbb{R}^r,\mathcal{B}_r)$ iteratively by \ref{thm:product-measure-2026a} (probability measures are $\sigma$-finite). Call a function $\varphi:\mathbb{R}^r\to\mathbb{R}$ \textbf{jointly Borel} if it is \reftext{def:measurable-function-2026a}{measurable} from $(\mathbb{R}^r,\mathcal{B}_r)$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space} and let $V_1,\dots,V_r$ be \reftext{def:independence-events-rvs-2026a}{independent} random variables on it with \reftext{def:distribution-cdf-random-variable-2026a}{distributions} $\nu_1,\dots,\nu_r$. Then the following hold.

\textbf{Claim 1.} The map $V:\Omega\to\mathbb{R}^r$ given by $V(\omega)=(V_1(\omega),\dots,V_r(\omega))$ is measurable from $(\Omega,\mathcal{F})$ to $(\mathbb{R}^r,\mathcal{B}_r)$, and its distribution $P_V:\mathcal{B}_r\to[0,1]$, $P_V(C)=P(V\in C)$, is a probability measure equal to $\nu_1\otimes\cdots\otimes\nu_r$.

\textbf{Claim 2.} If $\varphi:\mathbb{R}^r\to\mathbb{R}$ is jointly Borel and either nonnegative or bounded, then $\varphi(V_1,\dots,V_r)=\varphi\circ V$ is a random variable and its \reftext{def:expectation-variance-2026a}{expectation} is defined (as an element of $[0,\infty]$ in the nonnegative case, and as a real number in the bounded case, where $\varphi\circ V$ is \reftext{def:lebesgue-integral-integrable-2026a}{integrable}), with

In particular this expectation depends only on $\varphi$ and the distributions $\nu_1,\dots,\nu_r$.

\textbf{Claim 3 (block independence).} Let $I=\{i_1<\dots<i_p\}$ and $J=\{j_1<\dots<j_q\}$ be disjoint nonempty subsets of $\{1,\dots,r\}$, and let $\varphi:\mathbb{R}^p\to\mathbb{R}$ and $\psi:\mathbb{R}^q\to\mathbb{R}$ be jointly Borel. Then $\varphi(V_{i_1},\dots,V_{i_p})$ and $\psi(V_{j_1},\dots,V_{j_q})$ are independent random variables.

\textbf{Claim 4.} Each coordinate projection $(x_1,\dots,x_r)\mapsto x_i$ and the addition map $(x_1,\dots,x_r)\mapsto x_1+\cdots+x_r$ are jointly Borel; moreover, if $\varphi:\mathbb{R}^r\to\mathbb{R}$ is jointly Borel and $t:\mathbb{R}\to\mathbb{R}$ is Borel measurable, then $t\circ\varphi$ is jointly Borel.