## A question about a lemma in Kenneth Kunen’s article “Some points in $\beta \mathb{N}$.”

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!!

Posted on Categories proxies