Proof of Existence of Independent and Identically Distributed Sequences

Apply Existence of Independent Sequences with Prescribed Distributions to the constant sequence $\nu_m=\nu$ for every $m\in\mathbb{N}$ (with $\mathbb{N}$ the natural numbers). This produces exactly the probability space named in the statement — $\Omega=(0,1)$, $\mathcal{F}$ the Borel subsets of $(0,1)$, and $P$ the restriction of Lebesgue measure, a probability measure since $(0,1)$ has measure $1$ — together with an independent sequence $(X_m)_{m\in\mathbb{N}}$ of random variables on it such that every $X_m$ has distribution $\nu$. Since all the $X_m$ share the distribution $\nu$, the sequence is independent and identically distributed with common distribution $\nu$, as required. $\blacksquare$