Existence of Independent and Identically Distributed Sequences

Let $\nu$ be a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, with the Borel $\sigma$-algebra. Then there exist a probability space $(\Omega,\mathcal{F},P)$ and a sequence $(X_m)_{m\in\mathbb{N}}$ of random variables on it that is independent and identically distributed with common 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 Lebesgue measure, which is a probability measure since the interval $(0,1)$ has Lebesgue measure $1$.