Given an increasing sequence of finite sigma algebras $\mathcal{F}_n$ on the measurable space $(\Omega,\mathcal{F})$ and a consistent sequence of probabilities $P_n$ such that $P_{n+1}$ restricted to $\mathcal{F}_n$ is the same as $P_n$, there exists a probability $P$ on $\sigma(\bigcup_{n=1}^{\infty} \mathcal{F}_n)$ such that $P\big|_{\mathcal{F}_n}=P_n$
Where can I find a proof of this statement? What tools do I need to prove it?