Measure theory MOC

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 ∈Σ is said to be inner regular iff

𝜇(𝐴)=sup{𝜇(𝐾):𝐾⊆𝐴,𝐾∈Σ∩K}

and outer regular iff

𝜇(𝐴)=inf{𝜇(𝐺):𝐴⊆𝐺,𝐺∈Σ∩T}

A measure is called inner regular iff every measurable set 𝐴 ∈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. measure Thus a measure is regular iff

𝜇(𝐴)=sup{𝜇(𝐾):𝐾⊆𝐴,𝐾∈Σ∩K}=inf{𝜇(𝐺):𝐴⊆𝐺,𝐺∈Σ∩T}

for every 𝐴 ∈Σ.

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