I am studying the book Probability with Martingales by D Williams and I am confused with section 1.7. In this section, Williams introduces Caratheodory's Extension theorem. Here he states,
Let S be a set, Let $\Sigma_0$ be an algebra on S, and let $$\Sigma := \sigma(\Sigma_0).$$ If $\mu_0$ is a countably additive map $\mu_0 : \Sigma_0 \to [0,\infty]$, then there exists a measure $\mu$ on $(S,\Sigma)$ such that $$\mu=\mu_0$$ on $\Sigma_0$. If $\mu_0 < \infty$, then, by Lemma 1.6, this extension is unique - an algebra is a $\pi$-system!
I do not understand the conclusion of the theorem, is it not always the case that an algebra is a $\pi$-system? An algebra is closed under finite intersections and this perfectly matches the definition of a $\pi$-system. Why is the conclusion to the theorem so surprising as to warrant an exclamation?
Many thanks
As far as the exclamation is concerned, I think all one can say is that Williams is a...colorful writer (sometimes to his detriment, IMHO).
The bit about an algebra being a $\pi$ system is not part of the conclusion of the result (since, as you point out, this is trivially true by the definitions), but rather part of the result's proof. Using Lemma 1.6 (which is about $\pi$ systems) and the fact that an algebra is a $\pi$ system, one can conclude the stated uniqueness result.