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Separability of the L1-Space of a Vector Measure

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Separability of the L1-Space of a Vector Measure SEPARABILITY OF THE L'-SPACE OF A VECTOR MEASURE by WERNER J. RICKER (Received 22 June, 1990) Let 2 be a a-algebra of subsets of some set Q and let n :2—»[0, °°] be a a-additive measure. If 2(j*) denotes the set of all elements of 2 with finite jU-measure (where sets equal [i-a.e. are identified in the usual way), then a metric d can be defined in 2(ji) by the formula d(E, F) = n{E AF) =\\XE- XF\ dp (£, F e 2); (1) here £ AF = (E\F) U (F\E) denotes the symmetric difference of E and F. The measure ^ is called separable whenever the metric space (2(^t), d) is separable. It is a classical result that ju is separable if and only if the Banach space L'(ju) is separable [8, p. 137]. To exhibit non-separable measures is not a problem; see [8, p. 70], for example. If 2 happens to be the a-algebra of /z-measurable sets constructed (via outer-measure fi*) by extending H, defined originally on merely a semi-ring of sets Fcl, then it is also classical that the countability of T guarantees the separability of fi and hence, also of Ll(p), [8, p. 69]. There arises the natural question of what form such classical results on separability of Ll-spaces should take for vector-valued measures. We aim to formulate such results in this note. So, suppose that A' is a locally convex space (briefly, lcs), always assumed to be Hausdorff and sequentially complete. A a-additive map m:2—*X, where 2 is a a-algebra of subsets of some set Q, is called a (A*-valued) vector measure. A 2-measurable function /: Q —» C is called m-integrable if it is integrable with respect to the complex measure (m,x') :£>-» (m(E),x'), for £e2, for every x' e X' (the continuous dual space of X), and if, for every £e2, there exists an element of X, denoted by J"Efdm, which satisfies (JEfdm,x') = JEfd(m,x'), for every x' e A". The linear space of all m-integrable functions is denoted by L{m). Let 2.x denote the family of all continuous seminorms in X or, at least enough seminorms to determine the topology of X. Each q e 3.x induces a seminorm q{m) in L(m) via the formula (feL(m)), (2) where if^ c A" denotes the polar of the unit ball Uq = g~'([0,1]). The seminorms (2), as q varies through 3.x, define a lc topology r(m) in L(m). Since r(m) may not be Hausdorff we form the usual quotient space of L(m) with respect to the closed subspace Pi <7~'({0}). The resulting Hausdorff space (with topology again denoted by t(m)) is denoted by L\m); it can be identified with equivalence classes of functions from L(m) modulo /n-null functions, where a function / e L(m) is m-null whenever $Efdin = 0, for every £ e 2. All of the above definitions and further properties of Ll(m) can be found in [6]. Let 2(m) denote the subset of L\m) corresponding to {%E\ E 6 2} c L(m). Of course, elements of 2(m) can also (and will) be identified with equivalence classes of elements from 2. The formula (1) suggests how to topologize 2(m). Namely, we restrict Glasgow Math. J. 34 (1992) 1-9. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 26 Sep 2021 at 09:25:16, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500008478 2 WERNER J. RICKER 1 the L (m)-topology r(m) to 2(m). That is, each seminorm q(m), where q e SLX, induces a semi-metric dq on 2(w) by the formula dq{xE,XF) = q(m)(xE-XF) (E,Fel). (3) Again r(/n) will denote the uniform structure and topology in 2(m) so defined by the semi-metrics (3) as q varies through 2tx- 1. Main results. Throughout this section X is a Hausdorff, sequentially complete lcs. A vector measure m:2-»A" is called separable whenever the topological space (2(/n), r(m)) is separable. For X = C (or R) this coincides with the classical definition. PROPOSITION 1. Let m :H—>X be a vector measure. (i) If the measure m is separable, then the lcs Ll{m) is separable. (ii) Let the lcs X be metrizable. Then m is separable if and only if L\m) is separable. Proof, (i) Let B c 2(m) be a countable r(m)-dense set in 2(m). Then the collection * a £f(B) of all simple functions of the form E <XJXF(J) for k positive integer, a-, a "rational complex number" and F(j)eB, l</<&, is also countable. By the r(/n)-density of the n 2-simple functions in L (m), [6, Ch.2], it suffices to show that if / = E PJXEU) is a 2-simple function and positive numbers er are given together with seminorms qre3.x, l<r^k, then there exists an element h e &"{B) satisfying qr{m){f-h)<er (lsrsk). (4) Let e = min{er;l<r<A:} and K = max{qr(m)(xn); lsr<*}. Choose "rational complex numbers" a;-,ls/<n, satisfying |a)r- /3,| < e/(2nK) for l</<n. By r(m)- density of B in 2(m) there exist sets F(y) e B such that, for every ; e {1,2,. , n} we have dq£E(j), F(j)) = qr{m){xE(i) ~ XFU)) < ej/(2np), (5) for every l<r<&, where j8 = max{|j8y|; 1 </<«}. Let h be the element E (XJXFU) °f 5^(5). Since /=1 n n a \f ~ ^\ — 2J \ j~ Pj\ XF(J) + 2-1 10/1 • IZF(/) ~ #£(/)l /=1 y=l it follows that 1 \f-h\* e{2nK)- 1 ^(>) + p t |^0)-^(/)|. (6) Since ^(w)(^0))<^(m)(^n), for every l</<n and l<r</:, and q(m)(g) = i(m)(\8\)> for everY QZ&x and geL\m)—see (2)—it follows from (5), (6) and the definitions of K and e that (4) is satisfied. (ii) If X is metrizable, then 3.x can be chosen to be a countable set. It is then clear from the definition of x{m) that L\m) is also a metrizable lcs. Since (2(m), r(m)) is a subset of Ll{m) with the relative topology it follows that (2(m), r(m)) is separable whenever Lx{m) is separable [8, p. 20]. • Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 26 Sep 2021 at 09:25:16, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500008478 THE L'-SPACE OF A VECTOR MEASURE 3 It would seem useful to have available a criterion for determining separability. Given a measure m :2—»X we recall that 2 is called m-essentially countably generated [6, p. 32] if there exists a countably generated a-algebra 20c2 such that 2(m) = 20(m). PROPOSITION 2. Let m:'Z—>X be a vector measure. If 2 is m-essentially countably generated, then m is a separable measure. In particular, Ll(m) is separable. The proof of this result relies on the following two facts: the first is straightforward and the second follows from the first and [3, III Lemma 8.4]. LEMMA 1. (i) Let A be a family of subsets of a set Q. Then the a-algebras of subsets of Q generated by A U {Q} and by A coincide. (ii) Let 2 be a countably generated a-algebra of subsets of a set Q. Then there exists a countable algebra of sets Iocl such that the a-algebra generated by 20 is precisely 2. Proof of Proposition 2. Let 2,, be a countable algebra of subsets of Q which m-essentially generates 2. Let m0 denote the restriction of m to 20. Then m0 is a-additive on 20 and has an extension to a a-additive measure on 2, namely m. It follows from the equivalence of (i) and (xi) in the Theorem of Extension in [5] (the topology r*(ra) stated there in (xi) coincides with our r(m); see p. 178 of [5]) that 20 is r(m)-dense in 2 — 2(m). Accordingly, m is separable. D COROLLARY 1. Let m:^Z—*X be a vector measure. If 2 is m-essentially countably generated, then the closed subspace of X generated by the range of m is separable for the relative topology induced by X. Proof. The integration map <& given by <&:/'-» Ja/d/n, for f eLl(m), is continuous from (Ll(m), r(m)) into X. Let Y denote the closed subspace of X generated by the range, m(2) = {m(E); E €2}, of m. By approximating elements of Ll(m) by 2-simple functions it is clear that <J>(L'(m))cy and hence, the closure $(L'(m))cy. But, the formula m(E) = *(^E), for E e 2, shows that actually 4>(L'(m)) = Y. The proof of Proposition 2 showed that there exists a countable algebra of sets B which m-essentially generates 2 and such that B is r(m)-dense in 2(m). Then the collection 5^(B) of "rational" B-simple functions as denned in the proof of Proposition l(i) is countable and dense in L\m).
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