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