The article reference is: Kunen, K. (1976). Some points in βN. Mathematical Proceedings of the Cambridge Philosophical Society, 80(3), 385-398. doi:10.1017/S0305004100053032.

I am stuck in lemma 5.2 which says: Let $ \mathcal{U}$ be a selective ultrafilter and $ (\mathcal{M}, v)$ a non-atomic measure algebra. Then, in $ V^{\mathcal{M}}$ , there is no $ \mathcal{P}$ -point extending $ \mathcal{U}$ . The proof goes like this:

“Define a finitely additive measure $ \rho$ on $ \mathcal{P}(\omega)$ in $ V^{\mathcal{M}}$ as follows. If $ [[x\subseteq \omega]]=1$ , define a measure $ \sigma_x$ on $ \mathcal{M}$ (in $ \mathcal{V}$ ) by

$ \sigma_x(b)=\mathcal{U}- \lim (v([[n\in x]] \wedge b):n\in \omega)$ .

$ \sigma_x$ may be identified with an $ \mathcal{M}$ -valued element of $ [0,1]$ , which we call $ \rho(x)$ . “

(The following is the assertion which i’m troubling with:)

“Since $ \mathcal{M}$ is non-atomic, $ \rho$ is with value 1 non-atomic. “

Why is $ \rho$ with value 1, non-atomic?

Sorry if the question is to little elaborate, but is something very specific I need to understand. Thanks for the help!!