Let $X=R_d \times R$, where $R_d$ denotes the set of real numbers with the discrete topology and $R$ the set of real numbers with the natural topology. For every $f \in C_c(X)$, one has $f(\{x\} \times R ) \neq 0$ for at most a finite number of $x_1, ...,x_m \in R$. We put $I(f)=\sum_{i=1}^m \int_{R} f(x_i, y)\text{d}y$. Then $I$ is a positive linear functional on $X$. Let $\mu$ be the measure corresponding to $I$ by the Riesz representation theorem.
How to show that $\mu(R_d \times \{0\})=\infty$ (or more generally, $\mu(A \times \{0\})=\infty$ if $A$ is not countable)?
Thanks.