Is it true that:
"given a random variable $Z_i$ taking value in the Borel space $(\mathcal{Z}_i, \mathcal{B}_{\mathcal{Z}_i})$, there exists a function $l_i:[0,1]\rightarrow \mathcal{Z}_i$ such that $Z_i=l_i(\xi_i)$ with $\xi_i\overset{d}{\sim} U([0,1])$. Moreover, $l_i$ is not unique."
Is this result obvious? Which assumption plays a crucial role?
Reading some sources, is instead the correct result stating that $Z_i\sim l_i(\xi_i)$ in place of $Z_i= l_i(\xi_i)$ where $\sim$ denotes equivalence in distribution?