On a representation of random variables

Let $\xi$ and $\eta$ be arbitrary random variables. It is proved that there exists an independent of $\eta$ random variable $\zeta$, such that $\xi$ is a function of $\eta$ and $\zeta$.
This result is applied to prove the existence, for any $\delta>0$, of a $\delta$-anticipating strong solution of an Itô stochastic equation with bounded drift and unit diffusion coefficient.