Let $(\nu_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence} of probability \reftext{def:measure-measure-space-2026a}{measures} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, where $\mathbb{R}$ is the set of \reftext{def:real-numbers-c54-2026c}{real numbers}, $\mathcal{B}(\mathbb{R})$ is the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra}, and $\mathbb{N}$ is the set of \reftext{def:natural-numbers-2026a}{natural numbers}. Then there exist a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$ and a sequence $(X_m)_{m\in\mathbb{N}}$ of random variables on it that is \reftext{def:independence-events-rvs-2026a}{independent} and such that $X_m$ has \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $\nu_m$ for every $m$.

One may take $\Omega=(0,1)$, $\mathcal{F}=\{B\in\mathcal{B}(\mathbb{R}):B\subseteq(0,1)\}$, and $P$ the restriction to $\mathcal{F}$ of \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure}, as in \ref{thm:existence-iid-sequence-2026a}; taking all $\nu_m$ equal to a fixed probability measure $\nu$ recovers the statement of that theorem.