Problem: Define the distance function between $(x_1, y_1)$ and $(x_2, y_2)$ for two points (where $x_i , y_i$ are real numbers) $d((x_1, y_1), (x_2, y_2))$ in the plane to be $$|y_1 - y_2| \text{ if } x_1=x_2; \quad 1 + |y_1- y_2| \text{ if } x_1 \ne x_2.$$
$d$ is indeed a metric, and $(X, d)$ is a Locally compact Hausdorff space. Also, for any $f \in C_c(X)$, there's only finitely many $x_i$ such that there exists a $y$ with $f(x_i, y)\ne 0$, denoted $(x_1, \dots, x_n)$. (These serve as previous subparts of this problem, which are resolved).
For $f$ above define $I(f) = \sum_{i=1}^{n}\int_{-\infty}^{\infty} f(x_i, y) dy$. Then $I(f)$ induces a Random measure $\mu$ on $X$. Is $\mu$ inner regular for all Borel set?
My effort: I don't know what are the tools for proving inner regularity on Borel measures. On Folland, we know that every $\sigma$-finite Radon measure is inner regular. Also, based on my search, Radon measure is inner regular iff it is semi-finite. I think $X$ is not $\sigma$-finite, since a compact set in $X$ can only consists of finitely many $x$, and I don't know if it is semi-finite or not.