Regular measure

Let $𝑋$ be a topological space $(𝑋, T)$ and a measure space $(𝑋, Σ, 𝜇)$ . Let $K$ denote the set of all compact subsets of $𝑋$ . A measurable set $𝐴 \in Σ$ is said to be inner regular iff

𝜇 (𝐴) = s u p {𝜇 (𝐾) : 𝐾 \subseteq 𝐴, 𝐾 \in Σ \cap K}

and outer regular iff

𝜇 (𝐴) = i n f {𝜇 (𝐺) : 𝐴 \subseteq 𝐺, 𝐺 \in Σ \cap T}

A measure is called inner regular iff every measurable set $𝐴 \in Σ$ is inner regular, and likewise a measure is called outer regular iff every measurable set is outer regular. A measure which is both inner regular and outer regular is called regular. #m/def/measure Thus a measure is regular iff

𝜇 (𝐴) = s u p {𝜇 (𝐾) : 𝐾 \subseteq 𝐴, 𝐾 \in Σ \cap K} = i n f {𝜇 (𝐺) : 𝐴 \subseteq 𝐺, 𝐺 \in Σ \cap T}

for every $𝐴 \in Σ$ .

#state/tidy | #lang/en | #SemBr