Let M= collection of all borel measurable subsets Y of X such that
μ(Y)=sup{μ(K)∣K⊂X,K closed}=inf{μ(U)∣U⊂X,U open}. We will show M is σ− algebra containing all closed sets and hence all open sets. This will show μ is regular.
Now let Y∈M. Then, for ϵ>0, there is U open and K closed such that, μ(U)<μ(Y)+ϵμ(K)>μ(Y)−ϵ. Hence μ(Yc)>μ(Uc)+ϵ and μ(Yc)<μ(Kc)−ϵ Hence M is closed under complementation. Now let Yi be a countable collection. Then there exists Ui open and Ki closed such that μ(Ki)>μ(Yi)−ϵ/2i and μ(Ui)<μ(Yi)+ϵ/2i Let K=∩Ki and U=∪Ui . Then since its given the measure is σ finite, μ(U)=limn→∞μ(Ui)<μ(Y)+Σi=1∞ ϵ/2i =μ(Y)+ϵ. Similarly μ(K)>μ(Y)−ϵ. Hence M is closed under countable union. Now we have to show that it contains all closed sets. Let Y be closed. Let Un={x∈X∣d(y,x)<1/n for some y∈Y}. Then Un open and U1⊃U2⊃U3⋯ Also as Y is closed, so for any x∈/Y d(x,Y)=m>0.⇒m>1/n for some n. Hence ∩iUi=Y Hence μ(Y)=lim μ(Un)= inf μ(Un) ≥inf μ(U) ∀ U⊃Y. . But RHS ≥μ(Y) and hence the equality. For the other side we note Yis a closed set and hence is the supremum. Hence all closed sets are in M. Hence done.
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