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I was wondering what relations are between these different definitions of a regular measure? When are they equivalent?

There are two non-equivalent definitions from Wikipedia

Let $(X, T)$ be a topological space and let $\mathcal B$ be its Borel $σ$-algebra on $X$. Let $μ$ be a measure on $(X, Σ)$.

$\mu$ is regular, if $\forall A \in \mathcal B$ $$ \mu (A) = \sup \{ \mu (F) | F \subseteq A, F \mbox{ closed} \} $$ and $$ \mu (A) = \inf \{ \mu (G) | G \supseteq A, G \mbox{ open} \}. $$

Alternatively, $\mu$ is regular, if $\forall A \in \mathcal B$ and $\forall δ > 0$, there exists a closed set $F$ and an open set $G$ such that $$ F \subseteq A \subseteq G $$ and $$ \mu (G \setminus F) < \delta. $$

The two definitions are equivalent if $\mu$ is finite (otherwise, the second definition is stronger).

Some authors (such as Dudley and Royden) require the set $F$ to be compact instead of closed. From Royden's Real Analysis

Let $\mu$ be a measure defined on a $\sigma$-algebra $\mathcal M$ of subsets of $X$ and suppose that $\mathcal M$ contains the Baire sets. $\mu$ is said to be regular, if $\forall E \in \mathcal M$ $$\mu(E) = \inf \{\mu(O) | E \subseteq O, O \text{ open}, O \in \mathcal M\}.$$ $$\mu(E) = \sup \{\mu(K) | K \subseteq E, K \text{ compact}, K \in \mathcal M\}.$$

From encyclopediaofmath

On a Borel sigma algebra $\mathcal B$, a measure $\mu$ is called regular, if $$\forall A \in \mathcal B, \mu(A) = \sup \{\mu(F) | F \subseteq A, F \text{ closed} \}$$

Thanks and regards!

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