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On Supports of Regular Borel Measures Pacific Journal of Mathematics ON SUPPORTS OF REGULAR BOREL MEASURES DELMA JOSEPH HEBERT,JR. AND HOWARD E. LACEY Vol. 27, No. 1 January 1968 PACIFIC JOURNAL OF MATHEMATICS Vol. 27, No. 1, 1968 ON SUPPORTS OF REGULAR BOREL MEASURES D. J. HEBERT AND H. ELTON LACEY The existence of a regular Borel measure whose support is a given compact Hausdorff space X imposes definite struc- tures on X, C(X), and C(X)*. In this paper a necessary and sufficient condition is given to insure that X is the support of a regular Borel measure. This involves the intersection number of a collection of open sets in X. Measures which vanish on a sigma ideal of a sigma field of subsets of X which contains a basis for the topology of X are also considered. In particular, for a certain class of compact Hausdorff spacs X, necessary and sufficient conditions are given to insure the existence of a nonatomic regular Borel measure whose support is X. The final section of the paper is devoted to a study of normal measures; i.e., measures which vanish on meager Borel sets. Normal measures on X are shown to be related to normal measures on the projective resolution of X. NOTATION AND TERMINOLOGY. Set theoretical and topological terminology is that of [12], the terminology of linear topological spaces is that of [14], and measure theory terminology follows [11]. All spaces considered are taken to be nonempty and all measures considered are finite. If X is a compact Hausdorff space, C(X) denotes the space of continuous real-valued functions on X in the supremum norm, C(X)* denotes the space of all continuous linear functionals on C(X), or, equivalently, the space of all signed regular Borel measures, and B(X) denotes the space of all bounded real-valued functions on X in the supremum norm. 1* Intersection numbers* The following definitions are moti- vated by the concept of an intersection number as given in [13]. Let X be a compact Hausdorff space and B be a Boolean algebra. 1.1. If S = (/i, •••,/„) is a finite sequence in B(X),ί(S) = (l/tt)IIΣΓ=i/ίl|. If ASC(I), then I(A) = mf {i(S): S is a finite sequence in A}. 1.2. If S = (Au •••, An) is a finite sequence of subsets of X, i(S) = m&x{(k/n): there is a subsequence (Ail9 , Aik) of S such that fYJ=1Ai9 Φ0}. If H is a collection of subsets of X, then I(H) — inf {i(S): S is a finite sequence in H}. 1.3. If S = (Eu , En) is a finite sequence in B, then i(S) = 101 102 D. J. HEBERT AND H. ELTON LACEY ma,x{(k/n): there is a subsequence (Eil9 , Ein) of S such that and I(H) = inf {ΐ(S): S is a finite sequence in H). The relationship between the above concepts is the following: if Y is the Stone space of B (see [10] or [18]), then for HQB, I(H) = I{h(E): Ee H) = I{Ch{E): Ee H}, where h is the isomorphism of B onto the clopen (i.e., closed and open) sets of Y and Ch{E) is the charac- teristic function of h(E) (this notation for the characteristic function is used throughout). The numbers I(A) and I(H) above are called the intersection numbers of the collections A and H respectively. LEMMA 1.4. Let X be a compact Hausdorff space and let F and G be nonempty subsets of the positive cone of C(X). If a > 0 is such that, for each feF, there is a geG with a g tί f, then a I(G) £ I(F). Proof. Let S = (fl9 ,/Λ) be a finite sequence in F. For each i, 1 ^ ί ^ n, there is a gi in G such that ag{ ^ fim Thus \ and Σ/i If T=(glt •••,&,), then n α/(G) g αΐ(Γ) = a(~ n i —1 ϊ = l Hence α/(ί?) ^ DEFINITION 1.5. Let B be a Boolean algebra and let H be a nonempty subset of B. Then H is said to be positive if and only if I(H) > 0. Similarly, if H is a nonempty collection of nonempty subsets of a given set, then H is said to be a positive collection whenever 0. THEOREM 1.6. If X is a compact Hausdorff space with topology G, then there is a regular Borel measure whose support is X if and only if G\{ψ} is the union of a countable family of positive collections. (See [13], Th. 4) Proof. Suppose G\{φ} is the union of a countable family {GJ of positive collections. For each / in C(X) and each u, let A,= ON SUPPORTS OF REGULAR BOREL MEASURES 103 and Fn = {feC(X):f^ 0, ||/|| = 1, and Af is in Gn}, and denote the convex hull of Fn by Qn. It is a matter of computation to show that \\f\\^I(Fn) for each feQn and that for each feFn,{lj2)CAfSf. By Lemma 1.4, 0 < (l/2)/(GJ ^ I(Fn), and by the above, ||/||^J(JPw)>0 for all feQn. There is a positive linear functional ψn on C(X) such that ?„(/) ^ /(f ) for all /e Qn. For, if 17 = {g e C(X): \\g\\< I(Fn)}, then Qn + Z7 is open and convex and (Qn + U) Π — P = 0, where p= /eC(I)|/^0}. By [14], p. 118, there is a continuous linear functional φn on C(X) such that ψn(—f) ^ 0%(# + Λ) for each feP, g e Qn, and he U. Since 0 is in — P, 0 ^ 0n(# + /?,) for all g e Qn, and h e U. Thus φn is a positive linear functional and, without loss of 7 generality, it is assumed that \\φn\\ = 1. Suppose g e Q%, /(i ^) > ε > 0, and let 1 denote the constant function 1 on X. Then (s — I(Fn))l is in U, and 0 ^ ^ + (ε - I(Fn))l] = φn{g) - I{Fn)φn{l) + εφn(l) = pw(flf) - J(Fn) + ε and hence, φn(g) ^ I(Fn) - ε. Let ^> - Σ:«i (Wί* . Then ^ is a positive linear functional on C(X) and ||0|| = 1. Now, suppose FeP and f Φ 0. Then # = //||/|| has norm 1 and thus, g e Fn for some n. Hence, φ(g) ^ φn(g) ^ /(-Pn) > 0. It follows im- mediately that the regular Borel measure that corresponds to φ has support all of X. Conversely, suppose X has a regular Borel measure whose support is X. Then there is a positive normalized linear functional φ on X such that φ(f) > 0 if / ^ 0, / Φ 0. Let F = {/e C(X) | /^ 0, ||/|| = 1} and Fn = {feF\ φ(f) > (1/ri)}. By computation if follows that n For each feF and each n, let Bf = {x e X: f{x) > 0} and Hn = {J5y : feFn}. From Urysohn's lemma it follows that for each open subset J7 of X there is an feF such that f(x) — 0 for x$U and /(a) = 1 for some x e U and thus, f^CBfS Cv. If Gn = {TeG: U contains a member of ffw}, then for each U in Gn there is an feFn such that f^CB/^ CTJ. By Lemma 1.4, 0 < /(FJ ^ /(ίίj ^ /(GJ. Clearly G\{φ} = \J?=1Gn. COROLLARY 1.7. If X is a compact Hausdorff space and B is a basis for the topology G of X, then there is a regular Borel measure whose support is X if and only if B\{φ} is the union of A countable family of positive collections. 2. Nonatomic measures. In this section, certain conditions are shown to be sufficient for the existence of a nonatomic regular Borel measure whose support is a given compact Hausdorff space. The 104 D. J. HEBERT AND H. ELTON LACEY study of supports of nonatomic regular Borel measures is shown to be related to the study of perfect separable compact Hausdorff spaces. If m is a regular Borel measure on a compact Hausdorff space X, then m is called nonatomic if for each x in X, m({x}) = 0. PROPOSITION 2.1. [17] If X is a compact Hausdorff space, then there is a nonzero, nonatomic regular Borel measure on X if and only if X has a nonempty perfect subset. A well-known topological lemma is also needed. LEMMA 2.2. If C is a closed subset of a topological space X, then int C = int cl int C. THEOREM 2.3. If X is a perfect compact Hausdorff space such that there exists a regular Borel measure m on X whose support is X, then there are perfect subsets Xλ and X2 of X such that ( i ) either X1 is empty or there is a nonatomic regular Borel measure whose support is Xt, (ii) either X2 is empty or X2 is separable, (iii) X = X, U X2, (iv) intXi contains X\X2 and intX2 contains X\Xt. Proof. Suppose C = {xl9 x29 •} = {x in X: m({x}) > 0} and m = mγ + m2 where mx is nonatomic and m2 = Σ~=-i m({%n})eXn. Let Xλ = Supp (mO and let X2 = cl int Supp (m2). Since m^JS^Supp (mx)) == 0 and m2|X\Supp (m2)] = 0, if A = X\Supp (mx) Π X\Supp (m2), then 0 = m^A) = m2(A) = m(A) . Since the complement of the support of a regular Borel measure is open and since the m-measure of a nonempty open set is positive, A is open; therefore empty.
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