I have problems understanding the last part of a proof, concerning the existence of a measure on $\mathbb{Z}_p^\times$
Let $G=\text{Gal}(\mathbb{Q}(\zeta_{p^{\infty}})/\mathbb{Q})$ and define the Iwasawa algebra $\Lambda=\mathcal{O}[[G]]$, which is known to be isomorphic to the space of $\mathcal{O}$-valued measures of $G$ on the one hand and on the other hand to a product of $p-1$ copies of the ring of power series $\mathcal{O}[[T]]$, coming from the isomorphism $G\cong\mathbb{Z}_p^\times\cong\mathbb{F}_p^\times\times1+p\mathbb{Z}_p$. Further let $\chi:(\mathbb{Z}/p^m\mathbb{Z})^\times\rightarrow\overline{\mathbb{Q}}_p$ be a Dirichlet character and let $\kappa:G\overset{\sim}{\rightarrow}\mathbb{Z}_p^\times$ be the cyclotomic character.
In Motivic p-adic L-functions for families of modular forms, Lem.3 it is shown that there can't exist an element $\mu\in\Lambda$, such that $$ \int_G\chi^{-1}\kappa^nd\mu=\chi(-1), \text{ for all }n\in\mathbb{N}. $$ The proof uses the homomorphisms $\Phi_{n,\chi}:\Lambda\rightarrow\mathbb{Q}_p$ induced by $\chi^{-1}\kappa^n$ via the universal property of the Iwasawa algebra. Since $\overline{\mathbb{Q}}_p$ is a domain, each $\Phi_{n,\chi}$ must factor through some copy of $\mathcal{O}[[T]]$. Further, each character $\chi^{-1}\kappa^n$ determines a character on $\mathbb{F}_p^\times$, which is a power $\omega^{i(n,\chi)}$ of the Teichmüller character $\omega$, where $i(n,\chi)\in\{1,\dots,p-1\}$ depends on $\chi$ and $n$. This exponent does also determine the copy of $\mathcal{O}[[T]]$ over which $\Phi_{n,\chi}$ factorizes. Now let $P_{n,\chi}=\ker\Phi_{n,\chi}$. The space spec $\Lambda$ is the disjoint union of $p-1$ copies of spec $\mathcal{O}[[T]]$ and the copy over which $\Phi_{n,\chi}$ factorizes corresponds to the connected component in which $P_{n,\chi}$ lies. One can show easily that an infinite family of such $P_{n,\chi}$, which lies in spec $\mathcal{O}[[T]]$ is zariski dense([1], Lem. III.3.1 (b)).
For each $i\in\{1,\dots,p-1\}$ there are infinitely many pairs $(n,\chi)$ such that $i(n,\chi)=i$ and $\chi(-1)=1$, but one can also find infinitely many pairs such that $i(n,\chi)=i$ and $\chi(-1)=-1$. Therefore such a measure $\mu$ can't exist.
My problem is the last part, why does it follow that such a measure can't exist? I guess it should follow that the closure of the $P_{n,\chi}$ with $\chi(-1)=1$ does not contain the ones with $\chi(-1)=-1$ and the other way around. But it is not clear to me why this follows.