Here's the statement :
Let ($\Omega,\mathcal{F})$ and $(E,\mathcal{E})$ two measurable sets. Let $X:\Omega \to E$ a random variable $(\mathcal{F},\mathcal{E})-$measurable. Let $Y:\Omega\to \mathbb{R}$ a random variable $\sigma(X)-$measurable bounded (where $\sigma(X)=\{X^{-1}(B);B\in \mathcal{E}\}$). Then there exists $f:E\to \mathbb{R}$ which is $\mathcal{E}-$measurable bounded such that : $\forall \omega \in \Omega$, $Y(\omega)=f(X(\omega))$.
What are the applications of this theorem and what's the final point ?
Thanks in advance !
This theorem is useful to see that the conditional expaction $E[\xi|\eta]:=E[\xi|\sigma(\eta)]$ of a random variable $\xi$ given a $\sigma$-field generated by a random variable $\eta$ can be represented in the form $f(\eta)=E[\xi|\eta]$.
In fact, the random variable $E[\xi|\eta]$ is $\eta$-measurable, hence by the represantation theorm there exists a measurable mapping $f$, s.t. $E[\xi|\eta]=f(\eta)$.
Finally the notation $E[\xi|\eta=x]$ becomes clear: $E[\xi|\eta=x]=f(\eta(\omega))$ for an $\omega\in\Omega$ with $\eta(\omega)=x$