# Existence of a sequence of independent $E$-valued random variables with distribution $\mu$ given $\mu$ and $E$ Polish

I know that the following question is true for $$E=\mathbb{R}$$. I would like to know if it can be extended to Polish spaces.

Suppose that $$(E,d)$$ is a Polish space. Write $$\mathcal{B}(E)$$ for the Borel $$\sigma$$-algebra.
Let $$\mu:\mathcal{B}(E)\to [0,1]$$ be a probability measure.
It is true that there exists a sequence of independent $$E$$-valued random variables $$X_1,X_2,\ldots$$ define in some probability space $$(\Omega,\mathcal{F},P)$$ such that for any $$n$$, the distribution $$X_n$$ is precisely $$\mu$$, i.e. $$P\circ X_n^{-1}=\mu\quad\text{ for all }n\in\mathbb{N}$$ ? Any reference?

You can certainly do this. Define $$\Omega =E \times E \times \cdots$$. Let $$\mathcal F$$ be the sigma algebra generated by 'cylinder sets' [ i.e. sets of the type $$\omega: (w_{n_1} \in A_1,\cdots, w_{n_k} \in A_k)$$ with $$A_i$$'s Borel] and define $$P\{\omega: (w_{n_1} \in A_1,\cdots, w_{n_k} \in A_k)\}=\mu (A_1)\mu(A_2)\cdots \mu(A_k)$$. Extend $$P$$ to the Borel sigma algebra using Caratheodory extension Theorem. Define $$X_n(\omega)=\omega_n$$.