Existence of Independent Sequences with Prescribed Distributions

Let $(\nu_m)_{m\in\mathbb{N}}$ be a sequence of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, where $\mathbb{R}$ is the set of real numbers, $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra, and $\mathbb{N}$ is the set of natural numbers. 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 such that $X_m$ has 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 Lebesgue measure, as in Existence of Independent and Identically Distributed Sequences; taking all $\nu_m$ equal to a fixed probability measure $\nu$ recovers the statement of that theorem.