Define the set $X_1 = \{ x \in X: \exists y \in Y, (\phi^*)^{-1}\mu_x=\beta\mu_y+\mu,|\beta|>N,\mu\{y\}=0 \}.$
If $x \not\in X_1,$ what do I have?
Am I right to say that for any $y \in Y,$ we have $(\phi^*)^{-1}\mu_x=\beta\mu_y+\mu$ with $|\beta|\leq N$ and $\mu\{y\} \neq 0$?
More details:
$X,Y$ are locally compact spaces with first countability. $C(X)$ denotes complex-valued continuous functions on $X.$
Let $\phi:C(X) \rightarrow C(Y)$ be a linear isomorphism and norm-increasing with $\|\phi\| < 2.$ Its adjoint is denoted as $\phi^*:C(Y)^* \rightarrow C(X)^*.$
For any $x \in X,$ $\mu_x$ is a unit positive mass measure concentrated at $\{x\}$ only, that is, $\mu\{x \} =1$ and $\mu(X \setminus \{ x \}) = 0.$