Example.Given a measurable space $(\Omega ,{\mathcal{F}}_{\Omega })$, we can form the countable product ${\Omega }^{\mathbb{N}}$ and put a filtration on it viawhere ${\pi }_{\{1,\dots ,i\}}:{\Omega }^{\mathbb{N}}\to {\Omega }^i$ is the projection onto the first $i$ components

Proposition.Given a probability measure $P$ on $\Omega$, we can turn ${\Omega }^{\mathbb{N}}$ into a probability space with $P^{\mathbb{N}}$. Then the conditional probability of $x:\Omega \to X$ with respect to ${\mathcal{F}}_i$ is given by