Property of closed sets

Question

I am currently studying John Kingman's "Poisson Processes" and stumbled upon the following statements: "Every open set $G$ is a countable union of disjoint intervals $A_j$, and then $N(G) =$ $\sum N(A_j)$ is a random variable if the $N(A_j)$ are. (1)

Every closed set $F$ is the intersection of a decreasing sequence of open sets $G_i$, and $N(F)$ is the limit of the decreasing sequence of $N(G_i)$ ". (2)

I fail to understand why the first part of (2) is true, but I suppose the second part follows from the sigma-continuity of the $N(G_i)$, provided $N(F)$ is a random variable. I am grateful for any tip or piece of advice regarding the first part of (2), i.e. showing that every closed set $F$ is the intersection of a decreasing sequence of open sets.

Answer 1

This is true in any metric space. If $F$ is a closed set in a metric space then $F$ is the intersection of the sets $G_n=\{x: d(x,F) <\frac 1 n\}$ where $d(x,F)=\inf \{d(x,y): y \in F\}$. Also, each $G_n$ is open.

Answer 2

First part of (2):

Take $$ G_j = \{x \colon \inf_{y\in F} |x - y| < 1/j\} $$ [suppose we are working on $\mathbb R$].

Answer 3

You can take $$ F_{k} = \left\{ x+y \, : \, x \in F, \, |y| < \frac{1}{k} \right\}, $$ so that $F_k \supset F_{k+1}$ for all $k \geq 1$, and $$ \bigcap_{k=1}^\infty F_k = F. $$