Let $\nu$ be a probability \reftext{def:measure-measure-space-2026a}{measure} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, with the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra}. Then there exist a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$ and a \reftext{def:sequence-in-set-2026a}{sequence} $(X_m)_{m\in\mathbb{N}}$ of random variables on it that is \reftext{def:independence-events-rvs-2026a}{independent and identically distributed} with common \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $\nu$.

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}, which is a probability measure since the interval $(0,1)$ has Lebesgue measure $1$.