The question is from Folland Real Analysis Chapter 7 Exercise 12. Let $X=\mathbb{R}\times\mathbb{R}_{d}$,where $\mathbb{R}_{d}$ denotes $\mathbb{R}$ with the discrete topology. If $f$ is a function on $X$, let $f^{y}=f(x,y)$; and if $E\subset X$, let $E^{y}=\{x:(x,y)\in E\}$.
(b) Define a positive functional on $C_{c}(X)$ by $I(f)=\sum_{y\in R}\int f(x,y)dx$ and let $\mu$ be the associated Radon Measure on $X$. Then $\mu(E)=\infty$ for any $E$ such that $E^{y}\neq\emptyset$ for uncountable any $y$.
My thoughts are using the outer regularity of $\mu$ to proof that $\mu(U)=\infty$, where $U\in X$ is any open set contained $E$. Then by RRT, I think I should find $f\in C_c(X,[0,1])$ and $\mathrm{supp}(f)\in U$ such that $I(f)=\infty$ but I failed. I also found the post but still confused as I committed there. Any help would be appreciated.
