Regularity. Let X X be a metric space and be a finite measure on the

Regularity. Let $X$ be a metric space and $\mu$ be a finite measure on the Borel sets $B=B\left(X\right)$ and denote the open sets by $O$ and the closed sets by $F$. Define a family of sets

$$\Sigma :=\{A\subset X:\forall ϵ>0\exists U\in O,F\in F\text{s.t.}F\subset A\subset U,\mu (U\setminus F)<ϵ\}\text{.}$$

(i) Show that $A\in \Sigma ⟹A^c\in \Sigma$ and that $F\subset \Sigma$.

(ii) Show that $\Sigma$ is stable under finite intersections.

(iii) Show that $\Sigma$ is a $\sigma$-algebra containing the Borel sets $B$.

(iv) Conclude that $\mu$ is regular, i.e. for all Borel sets $B\in B$
$$\mu \left(B\right)=\underset{F\subset B,F\in F}{sup}\mu \left(F\right)=\underset{U\supset B,U\in O}{inf}\mu \left(U\right).$$
(v) Assume that there exists an increasing sequence of compact sets $K_j$ such that $K_j\uparrow X$. Show that $\mu$ satisfies
$$\mu \left(B\right)=\underset{K\subset B,K\text{compact}}{sup}\mu \left(K\right).$$
(vi) Extend the equality $\mu \left(B\right)={sup}_{F\subset B,F\in F}\mu \left(F\right)$ to a $\sigma$-finite measure $\mu$.