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 $\sigma(\bigcup_{k=1}^{\infty} \mathcal{F}_n)$ such that $P\big|_{\mathcal{F}_n}=P_n$
Why is $P$ well defined on $\cup_{n=1}^{\infty} \mathcal{F_k}$ I can easily show finite additivity of $P$ on $\cup_{n=1}^{\infty} \mathcal{F}_k$ using consistency above , but how can I show countable additivity?(In general I cannot use a large enough $m \in \mathbb{N}$ such that I use the countable additivity of the corresponding $P_m$ which is given to be a probability)
The author of the book I am reading (Ikeda and Watanabe) further claims that we can show that $P$ is indeed a probability measure on $\sigma(\cup_{k=1}^{\infty} \mathcal{F_k})$ using the Hopf's extension theorem. Can somebody give me a hint or some details?