(1) Show that the counting measure on $\Bbb Z$ with the induced metric from $\Bbb R$ is regular. (2) Show that the delta measure with respect to a point $x_0$ on any metric space is regular.
What I know is: A measure $\mu$ on a metric space with a given $\sigma$-algebra $M$ is called regular, if the $\sigma$-algebra of Borel sets is a subalgebra of $M$, and for every set $A$ in $M$, we have $\mu (A) = \inf \{ \mu (U) \; | \; U \text{ is open and } A \subset U \}$ and $\mu (A) = \sup \{ \mu (K) \; | \; K \text{ is compact and } K \subset A \}$.
Please give hints on both the parts on how to proceed.