Question #138323

If u be a borel measure on a metric space X and suppose there exists a sequence Gn of open sets, X=UnGn and for all n , u(Gn) < Infinity then u is regular.


1
Expert's answer
2020-10-15T18:35:47-0400



Let M= collection of all borel measurable subsets YY of X such that

μ(Y)=sup{μ(K)KX,K closed}=inf{μ(U)UX,U open}\mu(Y)=sup\{\mu(K)|K\subset X,K \ closed\}=inf\{\mu(U)|U\subset X,U \ open\}. We will show M is σ\sigma- algebra containing all closed sets and hence all open sets. This will show μ\mu is regular.

Now let YM.Y\in M. Then, for ϵ>0,\epsilon >0, there is UU open and KK closed such that, μ(U)<μ(Y)+ϵ\mu(U)<\mu(Y)+\epsilonμ(K)>μ(Y)ϵ\mu(K)>\mu(Y)-\epsilon. Hence μ(Yc)>μ(Uc)+ϵ\mu(Y^c)> \mu(U^c)+\epsilon and μ(Yc)<μ(Kc)ϵ\mu(Y^c)< \mu(K^c)-\epsilon Hence M is closed under complementation. Now let YiY_i be a countable collection. Then there exists UiU_i open and KiK_{i} closed such that μ(Ki)>μ(Yi)ϵ/2i\mu(K_i)>\mu(Y_i)-\epsilon /2^i and μ(Ui)<μ(Yi)+ϵ/2i\mu(U_i)<\mu(Y_i)+\epsilon /2^i Let K=KiK=\cap K_i and U=UiU=\cup U_i . Then since its given the measure is σ\sigma finite, μ(U)=limnμ(Ui)<μ(Y)+Σi=1 ϵ/2i\mu(U)=lim_{n\rightarrow \infty}\mu(U_i)< \mu(Y)+\Sigma_{i=1}^{\infty} \ \epsilon/2^i =μ(Y)+ϵ=\mu(Y)+\epsilon. Similarly μ(K)>μ(Y)ϵ.\mu(K)>\mu(Y)-\epsilon. Hence M is closed under countable union. Now we have to show that it contains all closed sets. Let YY be closed. Let Un={xXd(y,x)<1/n for some yY}.U_n=\{x\in X| d(y,x)<1/n \ \textrm{for some} \ y \in Y \}. Then UnU_n open and U1U2U3U_1\supset U_2\supset U_3\cdots Also as YY is closed, so for any xYx\notin Y d(x,Y)=m>0.m>1/nd(x,Y)=m>0. \Rightarrow m>1/n for some n.n. Hence iUi=Y\cap_i U_i =Y Hence μ(Y)=lim μ(Un)=\mu(Y)= lim \ \mu(U_n)= inf μ(Un)\mu(U_n) inf μ(U)  UY.\geq \textrm{inf} \ \mu(U) \ \forall \ U\supset Y. . But RHS μ(Y)\geq \mu(Y) and hence the equality. For the other side we note YYis a closed set and hence is the supremum. Hence all closed sets are in M. Hence done.


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