Regularity on Polish spaces. A Polish space X X is a complete metric space which has a

Regularity on Polish spaces. A Polish space $X$ is a complete metric space which has a countable dense subset $D\subset X$.

Let $\mu$ be a finite measure on $(X,B(X\left)\right)$. Then $\mu$ is regular in the sense that

$$\mu \left(B\right)=\underset{K\subset B,K\text{compact}}{sup}\mu \left(K\right)=\underset{U\supset B,U\text{open}}{inf}\mu \left(U\right)$$

[ use Problem 4.21 and show that the set $K:={\bigcap }_n{\bigcup }_{j=1}^{k\left(n\right)}{K}_{1/n}\left(d_{k\left(n\right)}\right)$ is compact; $K_{ϵ}\left(x\right)$ denotes a closed ball of radius $ϵ$ and centre $x$ and ${\left\{d_k\right\}}_k$ is an enumeration of $D$. Choosing $k\left(n\right)$ sufficiently large, we can achieve that $\mu \left(X\setminus K\right)<ϵ$.]

Data from problem 4.21

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

\[
\Sigma:=\{A \subset X: \forall \epsilon>0 \exists U \in \mathscr{O}, F \in \mathscr{F} \text { s.t. } F \subset A \subset U, \mu(U \backslash F)<\epsilon\} \text {. }
\]

(i) Show that \(A \in \Sigma \Longrightarrow A^{c} \in \Sigma\) and that \(\mathscr{F} \subset \Sigma\).

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

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