Let $(E_i,\mathcal E_i,\mu_i)$ be probability spaces for $i\in\{1,\dots,n\}$, and set \begin{align*} (E,\mathcal E,\mu)=(E_1\times\cdots\times E_n,\mathcal E_1\otimes\cdots\otimes\mathcal E_n,\mu_1\otimes\cdots\otimes\mu_n). \end{align*} Let $\pi_i:E\to E_i$ be the coordinate projection $\pi_i(x_1,\dots,x_n)=x_i$. Then $\pi_1,\dots,\pi_n$ are independent and $\mu_{\pi_i}=\mu_i$.