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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?

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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 $X_n(\omega)=\omega_n$.

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