Joint continuity of tensor products

J. Aust. Math. Soc. 74 (2003), 185–199

JOINT CONTINUITY OF INJECTIVE TENSOR PRODUCTS OF VECTOR MEASURES IN BANACH LATTICES JUN KAWABE

(Received 15 February 2001; revised 7 November 2001)

Communicated by A. Pryde

Abstract It is shown that the injective tensor product of positive vector measures in certain Banach lattices is jointly continuous with respect to the weak convergence of vector measures. This result is obtained by a diagonal convergence theorem for injective tensor integrals. Our approach to this problem is based on Bartle’s bilinear integration theory. 2000 Mathematics subject classification: primary 28A33, 28B05; secondary 46A11, 46A32. Keywords and phrases: positive vector measure, injective tensor product of vector measures, weak convergence of vector measures, Banach lattice, injective tensor integral.

1. Introduction In 1967, Duchon and Kluv´anek [8] introduced the notion of tensor products of vector measures which we describe in our setting: Let .; A / and .0; B / be measurable e Y the injective tensor product of X spaces. Let X and Y be Banach spaces, and X ⊗ and Y (see [7, Chapter VIII]). For any vector measures ¼ : A → X and ¹ : B → Y , e ¹ : A ×B → X ⊗ e Y , which is called there always exists a unique vector measure ¼ ⊗ e ¹/.A× B/ = ¼.A/⊗¹.B/ for all the injective tensor product of ¼ and ¹, such that .¼ ⊗ A ∈ A and B ∈ B . On the other hand, Dekiert [6] recently introduced the notion of weak convergence of Banach space-valued measures, which is a natural generalization of the weak convergence of probability measures, and studied its properties (see also [19, 22]). This work has been supported by Grant-in-Aid for General Scientific Research No. 11640160, the Ministry of Education, Science, Sports and Culture, Japan. c 2003 Australian Mathematical Society 1446-8107/03 $A2:00 + 0:00

185

186

Jun Kawabe

[2]

Let S and T be topological spaces. In this paper, we prove that if nets {¼Þ } and {¹Þ } of positive vector measures on S and T with values in Banach lattices X and Y e ¹Þ converge weakly to ¼ and ¹, respectively, then the injective tensor product ¼Þ ⊗ e ¹ under some additional assumptions. converges weakly to ¼ ⊗ The injective tensor product of two probability measures is just the usual product measure, so that its joint continuity is well known in the case that S and T are separable metric spaces [3, Theorem 3.2], and more generally completely regular spaces [24, Proposition I.4.1]. It was also shown in [5, Corollary to Theorem 1] that the convolution of probability measures on an arbitrary topological group is jointly continuous. In a recent paper [14], we studied this kind of problem for vector measures with values in certain nuclear spaces. It is shown in [14, Theorems 5 and 7] that the weak convergence of a net of injective tensor products of uniformly bounded nuclear space-valued vector measures follows from that of the corresponding net of real product measures. The way of proving the above result is essentially based on a finite dimensional feature of nuclear spaces, that is, the weak topology coincides with the original topology on every bounded subset of any barreled, quasi-complete nuclear space. Therefore, the same method may not apply to the case of vector measures with values in Banach spaces. The purpose of this paper is to show that this type of joint continuity of product measures remains true for the injective tensor products of positive vector measures in certain Banach lattices; see Theorem 5.4. Our approach to this problem is based on the Bartle bilinear vector integration [1]. In Section 2, we formulate some notation and results which are needed in the sequel. In Section 3, we give some technical results of the injective tensor integrals, which are the Bartle bilinear integrals with respect to the injective tensor product. In Section 4, we give a diagonal convergence theorem for the injective tensor integrals, which is not only crucial to prove our main result, but seems to be of some interest. Using this diagonal convergence theorem, it is shown in Section 5 that the joint continuity of the injective tensor products remains true for positive vector measures with values in Banach lattices under some additional assumptions. All the topological spaces and uniform spaces in this paper are Hausdorff, and the scalar fields of Banach spaces are taken to be the field R of all real numbers. Denote by N the set of all natural numbers.

2. Preliminaries Let X be a Banach space with norm k · k and X ∗ the topological dual of X. Let B X ∗ denote the closed unit ball of X ∗ . Let .; A / be a measurable space. A ¦ -additive set function ¼ : A → X is called a vector measure. The semivariation of ¼ is the set

[3]

Joint continuity of tensor products

187

function k¼k.A/ = sup{|x ∗ ¼|.A/ : x ∗ ∈ B X ∗ }, where |x ∗ ¼|.·/ is the total variation of x ∗ ¼. Then k¼k./ < ∞ [2, Lemma 2.2]. Let ¼ : A → X be a vector measure. Denote by A the indicator function of a set A. A ¼-null set is a set E ∈ A for which k¼k.E/ = 0; the term ¼-almost everywhere refers to the complement of a ¼-null set. Given an A -simple function P of the form f = R mk=1 ak Ak with a1 ; : : : ; am ∈ RR, A1 ; : : : ; A m ∈ A , m ∈ N, Pm define its integral A f d¼ over a set A ∈ A by A f d¼ = k=1 ak ¼.Ak ∩ A/. An A -measurable function f : → R is said to be ¼-integrable if there exists a sequence { f n } of R A -simple functions converging ¼-almost everywhere to f such that the sequence { f d¼} converges in the norm of X for each A ∈ A . This limit A n R f d¼ does not depend on the choice of such simple functions f n , n R∈ N. By A the Orlicz-Pettis theorem [7, Corollary I.4.4], the indefinite integral A 7→ A f d¼ is ¦ -additive.

Every bounded, A -measurable function f : → R is ¼-integrable, and R

f d¼ ≤ sup!∈ A | f .!/| · k¼k.A/ for each A ∈ A [2, Theorem 2.6]. For further A properties of this integral see [2, 10, 17, 18]. We define several notions of regularity for vector measures on a topological space. Let S be a topological space and B .S/ the ¦ -algebra of all Borel subsets of S. Let ¼ : B .S/ → X be a vector measure. We say that ¼ is Radon if for every " > 0 and A ∈ B .S/ there exists a compact subset K of A such that k¼k.A − K / < ", and it is tight if this condition is satisfied for A = S. We say that ¼ is − -smooth S if, for any increasing net {G Þ } of open subsets of S with G = Þ G Þ , we have limÞ k¼k.G − G Þ / = 0. It follows that ¼ is Radon (respectively tight, − -smooth) if and only if, for each x ∗ ∈ X ∗ , the real measure x ∗ ¼ is Radon (respectively tight, − -smooth). In fact, this is a consequence of the Rybakov theorem [7, Theorem IX.2.2], which ensures that there exists x0∗ ∈ X ∗ for which x0∗ ¼ and ¼ are mutually absolutely continuous. The following result can be proved as in the case of scalar measures; see for example [24, Proposition I.3.2]. PROPOSITION 2.1. Let S be a topological space and X a Banach space. Let ¼ : B .S/ → X be a − -smooth vector measure. Let { fÞ } be a uniformly bounded decreasing net of upper semicontinuous non-negative functions on S. If fR = lim Þ f Þ is R the pointwise limit of f Þ , then f Þ ’s and f are all ¼-integrable and limÞ S f Þ d¼ = f d¼. S 3. Some properties of injective tensor integrals In this section, we define the Bartle bilinear integration in our setting: Let X and e Y the injective tensor product of X and Y ; Y be Banach spaces. Denote by X ⊗ see [7, Chapter VIII]. Let .; A / be a measurable space. Let ' : → X be a vector

188

Jun Kawabe

[4]

function and ¹ : A → Y a vector measure. Given an X-valued simple function Pm ' = m ∈ X, A1 ; : : : ; A m ∈ A , m ∈ N, define its integral k=1 x k Ak with x 1 ; : : : ; xR R P e e d¹ = mk=1 x k ⊗ ¹.Ak ∩ A/. We say that ' is ' ⊗ d¹ over a set A ∈ A by A ' ⊗ A ¹-measurable if there exists a sequence {'n } of X-valued simple functions converging ¹-almost everywhere to '. The function ' is said to be ¹-integrable in the sense of Bartle if there exists a sequence {'n } of X-valued simple functions converging R e ⊗ d¹} converges in the norm ¹-almost everywhere to ' such that the sequence { ' A n R e e of X ⊗ Y for each A ∈ A . This limit A ' ⊗ d¹ does not depend on theRchoice of e d¹ such X-valued simple functions 'n ; n ∈ N, and the indefinite integral A → A ' ⊗ e Y -valued vector measure on A . is an X ⊗ For simplicity, weR say that the ' is ¹-integrable if it is ¹-integrable in the sense of e d¹ is called the injective tensor integral of ' over A with Bartle. The integral A ' ⊗ respect to ¹. See a recent paper [11] for further properties of injective tensor integrals such as some characterizations of integrable functions and the general Fubini theorem. In the following, we prepare some technical properties of injective tensor integrals which will be used in Section 4. The proof of the following lemma is obvious and will be omitted. LEMMA 3.1. Let .; A / be a measurable space. Let X and Y be Banach spaces. Let ¹ : A → Y be a vector measure and f : → R a ¹-integrable, A -measurable function. Then, given x ∈ X, the vector R function f x : →R X defined by . f x/.!/ = e d¹ = x ⊗ A f d¹ for each A ∈ A . f .!/x for ! ∈ is ¹-integrable and A . f x/ ⊗ When the Banach space X is equipped with the additional structure of a Banach lattice, we may introduce the notion of positivity for vector measures. We say that a vector measure ¼ : A → X is positive if ¼.A/ ≥ 0 for every A ∈ A . By [22, Lemma 1.1], for every positive vector measure ¼, we have k¼k.A/ = k¼.A/k for all A ∈ A . Further, it is easy to verify that for any ¼-integrable, A -measurable real functions f and g with | f | ≤ g ¼-almost everywhere, we have Z Z

Z

Z

Z

Z

f d¼ ≤ | f | d¼ ≤ g d¼ and f d¼ ≤ | f | d¼ ≤ g d¼ :

These facts greatly facilitate the analysis of positive vector measures and are used frequently without saying these explicitly in this paper. For further properties of positive vector measures on metric spaces, see [19, 22]. For other applications of positivity of vector measures, see [15]. We refer the reader to the book of Schaefer [20] for the basic theory of Banach lattices. PROPOSITION 3.2. Let .; A / be a measurable space. Let X be a Banach space and Y a Banach lattice. Let ¹ : A → Y be a positive vector measure. If an A -measurable function f : → R is ¹-integrable and a vector function ' :

[5]


189

→ X is bounded and ¹-measurable, then the vector function

R f ' : → X

. f '/ ⊗ e d¹ ≤ defined by . f '/.!/ = f .!/'.!/; ! ∈ , is ¹-integrable, and A

R

sup!∈ k'.!/k · A | f | d¹ for each A ∈ A . PROOF. The function ' is ¹-integrable by [1, Theorem 4], and hence we can find a sequence {'n } of X-valued simple functions on converging ¹-almost everywhere to ' such that k'n .!/k ≤ k'.!/k for all ! ∈ and n ∈ N (see, for instance, [25, Theorem 2.6]). Fix A ∈ A and n ∈ N and put k'k∞ = sup!∈ k'.!/k. The function f 'n which is ¹-integrable by Lemma 3.1 satisfies

Z

Z

∗

e (3.1)

. f 'n / ⊗ d¹ = sup f · .x 'n / d¹ ; x ∗ ∈ BX ∗

A

A

see [11, page 327]. Since | Rf .!/.x ∗ 'n /.!/| ≤ k'k∞ |Rf .!/| for all x ∗ ∈ B X ∗ , the positive measure ¹ satisfies A f · .x ∗ 'n / d¹ ≤ k'k∞ A | f | d¹, which implies that

Z

Z

∗

sup f · .x 'n / d¹ ≤ k'k∞ | f | d¹ (3.2)

: x ∗ ∈ BX ∗

A

A

Since f 'n converges ¹-almost everywhere to Rf ', it follows from R (3.1), (3.2) e d¹ → A . f '/ ⊗ e d¹. and [1, Theorem 10] that f ' is ¹-integrable and A . f 'n / ⊗ This together with (3.1) and (3.2) gives the required inequality. Let T be a topological space and B .T / the ¦ -algebra of all Borel subsets of T . Let C.T; X/ denotes the Banach space of all bounded continuous functions ' : T → X with the norm k'k∞ = supt ∈T k'.t/k. We write C.T / = C.T; R/. PROPOSITION 3.3 ([26, Theorem 1.6]). Let T be a topological space. Let X and Y be Banach spaces. Let ¹ : B

R.T / → Y be a tight vector measure and ' ∈ C.T; X/. e d¹ ≤ supt ∈ A k'.t/k·k¹k.A/ for all A ∈ B .T /. Then, ' is ¹-integrable, and A ' ⊗ PROOF. By [1, Theorem 4] we have only to prove that ' is ¹-measurable. Take an S∞ ∞ increasing sequence {K n }n=1 of compact subsets of T such that T − n=1 K n is ¹-null. S∞ Since '.K n / is compact for all n ∈ N, the set ' n=1 K n is ¦ -compact, and hence separable. Apply the Pettis measurability theorem [7, Theorem II.1.2] to conclude that ' is ¼-measurable. 4. A diagonal convergence theorem for injective tensor integrals Let T be a uniform space with the uniformity UT . Let X be a Banach space. Denote by U .T; X/ the Banach space of all bounded uniformly continuous functions

190

Jun Kawabe

[6]

' : T → X with the norm k'k∞ = supt ∈T k'.t/k. We write U .T / = U .T; R/. Let Y be a Banach lattice. Denote by M +.T; Y / the set of all positive vector measures ¹ : B .T / → Y , and denote by Mt+ .T; Y / the set of all ¹ ∈ M +.T; Y / which are tight. In this section, we give a diagonal convergence theorem for injective tensor integrals with respect to positive vector measures. The following theorem is not only crucial to prove our main results, that is Theorem 5.3 and Theorem 5.4, but seems to be of some interest. THEOREM 4.1. Let T be a uniform space with the uniformity UT . Let X be a Banach space and Y a Banach lattice. Consider a net {'Þ } ⊂ U .T; X/ and ' ∈ U .T; X/ satisfying the following conditions: .i/ 'Þ .t/ → '.t/ for every t ∈ T ; .ii/ {'Þ } is uniformly bounded, that is, supÞ k'Þ k∞ < ∞; and .iii/ {'Þ } is uniformly equicontinuous on T , that is, for any " > 0, there exists V ∈ UT such that supÞ k'Þ .t/ − 'Þ .t 0 /k < " whenever .t; t 0 / ∈ V .

+ Given a net R{¹Þ } ⊂ Mt+ .T; Y / and a − -smooth Rmeasure ¹ ∈ M R t .T; Y /, if R e d¹Þ = T ' ⊗ e d¹. limÞ T g d¹Þ = T g d¹ for every g ∈ U .T /, then limÞ T 'Þ ⊗

To prove Theorem 4.1, we need several auxiliary results. In what follows, for any V ∈ UT and t ∈ T , put V .t/ = {t 0 ∈ T : .t; t 0 / ∈ V }. LEMMA 4.2. Let T be a uniform space with the uniformity UT . Let Y be a Banach space. Assume that ¹ ∈ M .T; Y / is − -smooth. Then, for each " > 0 and V ∈ UT , we can find a finite subset {t1 ; t2 ; : : : ; tn } of T and h ∈ U .T / with 0 ≤ h ≤ 1 satisfying the following conditions: Sn .i/ h.t R i / = 0 for all i = 1; 2; : : : ; n; h.t/ = 1 for all t 6∈ i =1 V .ti /, and .ii/ T h d¹ < ". PROOF. Let V ∈ UT and fix " > 0. Then, for each a ∈ T , we can find da ∈ U .T / satisfying 0 ≤ da ≤ 1, da .a/ = 0, and da .t/ = 1 if t 6∈ V .a/ (the existence of such a function da follows from a proof of Uniformizable Theorem [13, Proposition 11.5]). Let Þ range over the finite subsets of T and for Þ = {t1 ; t2 ; : : : ; tn }, put h Þ .t/ = min1≤i ≤n dti .t/ for t ∈ T: Then, {h Þ } is a decreasing net in U .T / converging pointwise to 0 (the ordering for the Þ’s being inclusion). Since ¹ is − -smooth,

R the set-theoretical

by Proposition 2.1, we have T h Þ d¹ < " for some Þ = {t1 ; t2 ; : : : ; tn }, and this function h Þ is a required one. PROPOSITION 4.3. Let T be a uniform space with the uniformity UT . Let X be a Banach space and Y a Banach lattice. Let {¹Þ } be a net in M + .T; Y / and ¹ ∈ M + .T; Y /. Assume that ¹ is tight. Then the following two conditions are equivalent:

[7]


191

.i/ For every g ∈ U .T /, we have Z Z (4.1) g d¹Þ → g d¹: T

T

.ii/ For every ' ∈ U .T; X/, we have

Z

Z

∗ ∗

sup (4.2)

x ' d¹Þ − x ' d¹ → 0: x ∗ ∈ BX ∗

T

T

PROOF. It is routine to prove that (ii) implies (i), and hence we prove (i) implies (ii). Fix ' ∈ U .T; X/, and we may assume that k'k∞ ≤ 1 without loss of generality, since we obtain the result for a general ' by considering '=k'k∞ in place of '. Given " > 0, the tight measure ¹ satisfies k¹k.T − K / < " for some compact subset K of T . We first claim that there exists an open set V ∈ U T such that the inequality sup |u ∗ '.t/ − v∗ '.t/| ≤ 3"

(4.3)

t ∈K .V /

holds for all u ∗ ; v ∗ ∈ B X ∗ satisfying sup |u ∗ '.t/ − v∗ '.t/| < ";

(4.4)

t ∈K

where K .V / = {t ∈ T : .t; t 0 / ∈ V for some t 0 ∈ K }. In fact, there exists an open set V ∈ UT such that supx ∗ ∈ BX ∗ |x ∗'.t/ − x ∗ '.t 0 /| = k'.t/ − '.t 0 /k < " whenever .t; t 0 / ∈ V . Take elements u ∗ ; v ∗ ∈ B X ∗ such that (4.4) holds. Let t ∈ K .V / and take t 0 ∈ K satisfying .t; t 0 / ∈ V . Then, we have |u ∗ '.t/ − v∗ '.t/| ≤ |u∗ '.t/ − u∗ '.t 0 /| + |u ∗ '.t 0 / − v∗ '.t 0 /| + |v ∗ '.t 0 / − v∗ '.t/| < 3"; which implies (4.3). Next we claim that (4.5)

lim sup k¹Þ k.T − K .V // < ": Þ

Indeed, since K .V / is an open subset containing the compact set K , there exists g ∈ U .T / vanishing on K such that 0 ≤ g ≤ 1 and g = 1 on T −K .V / (adapt the proof of

R[13, Proposition

R11.5]). The positive measures ¹Þ and ¹ satisfy k¹Þ k.T − K .V // ≤

g d¹Þ and g d¹ ≤ k¹k.T − K /. It follows from assumption (i) applied T T

192

Jun Kawabe

[8]

R

R

to g ∈ U .T / that T g d¹Þ → T g d¹ . Hence, lim supÞ k¹Þ k.T − K .V // ≤ k¹k.T − K / < ", which establishes (4.5). Now, the set of functions {x ∗ ' : x ∗ ∈ B X ∗ } restricted to K is uniformly bounded and uniformly equicontinuous in C.K /, so that it is a relatively compact subset of C.K / by the Arzel`a-Ascoli theorem. In other words, there exists a finite subset B0 of B X ∗ such that for any x ∗ ∈ B X ∗ there is x0∗ ∈ B0 with supt ∈K |x ∗ '.t/ − x0∗ '.t/| < ", and hence we have supt ∈K .V / |x ∗'.t/ − x0∗ '.t/| ≤ 3" by (4.3). Let x ∗ ∈ B X ∗ and take x0∗ ∈ B0 as above. Then

Z

Z

∗ ∗

x ' d¹Þ − x ' d¹ (4.6)

T T

Z

Z

Z

∗ ∗ ∗ ∗

≤ |x ' − x 0 '| d¹Þ + max u ' d¹ − u ' d¹ Þ

u ∗ ∈B0 T T T

Z

∗ ∗

+

|x0 ' − x '| d¹ : T

Further, we have

Z

Z

∗ ∗ ∗ ∗

(4.7) |x ' − x 0 '| d¹Þ

|x ' − x 0 '| d¹Þ ≤

T

TZ−K .V /

+ |x ∗' − x 0∗ '| d¹Þ

K .V /

≤ 2k¹Þ k.T − K .V // + 3"k¹Þ .T /k and (4.8)

Z

∗ ∗

|x ' − x '| d¹ ≤ 2k¹k.T − K / + "k¹.T /k < 2" + "k¹.T /k: 0

T

It follows from (4.6)–(4.8) that

Z

Z

∗ ∗

sup x ' d¹Þ − x ' d¹ (4.9)

x ∗ ∈ BX ∗

T

T

≤ 2k¹Þ k.T − K .V // + 3"k¹Þ .T /k + 2" + "k¹.T /k

Z

Z

∗ ∗

+ max u ' d¹Þ − u ' d¹

: ∗ u ∈B0 T

T

R

R

Now an appeal to (4.1) gives k¹Þ .T /k = T 1 d¹Þ → T 1 d¹ = k¹.T /k and

Z

Z

∗ ∗

→ 0; max u ' d¹ − u ' d¹ Þ

u ∗ ∈B0 T

T

[9]


193

and hence by (4.5) and (4.9) we have

Z

Z

∗ ∗

lim sup sup x ' d¹Þ − x ' d¹

Þ

x ∗ ∈ BX ∗

T

T

≤ 2 lim sup k¹Þ k.T − K .V // + 3"k¹.T /k + 2" + "k¹.T /k Þ

< 4" + 4"k¹.T /k = 4".1 + k¹.T /k/; which establishes (4.2). PROPOSITION 4.4. Let T be a uniform space. Let X be a Banach space and Y a Banach lattice. Let {¹Þ } be a net in Mt+ .T; Y / and ¹ ∈ Mt+ .T; Y /. Then the following two conditions are equivalent: R R .i/ For every g ∈ U .T /, we have T gR d¹Þ → T g d¹. R e d¹Þ → T ' ⊗ e d¹. .ii/ For every ' ∈ U .T; X/, we have T ' ⊗ PROOF. It is obvious that (ii) implies (i), hence we (i)

implies (ii). R and R prove ∗ ∗

Fix ' ∈ U .T; X/. Then, supx ∗ ∈ BX ∗ T x ' d¹Þ − T x ' d¹ → 0 by Proposition 4.3. It follows from Proposition 3.3 that ' is integrable with respect to both ¹Þ and ¹. Hence, by [11, page 327] we have

Z Z

Z Z

∗ ∗ ∗ ∗

'⊗

e d¹Þ − ' ⊗ e d¹ = sup x 'd.y ¹Þ / − x 'd.y ¹/

T

T

x ∗ ∈ BX ∗ y ∗ ∈ BY ∗

T

T

Z

Z

∗ ∗

= sup x ' d¹Þ − x ' d¹

→ 0; x ∗∈ B ∗ X

T

T

which establishes (ii). PROOF OF THEOREM 4.1. Fix " > 0. By assumption (iii) we can find V ∈ UT such that (4.10)

sup k'Þ .t/ − 'Þ .t 0 /k < " Þ

and k'.t/ − '.t 0 /k < "

whenever .t; t 0 / ∈ V . Then, Proposition 3.3 allows linear R us to define continuous R e Y by L Þ = T ⊗ e d¹Þ and L = T ⊗ e d¹, operators L Þ ; L : U .T; X/ → X ⊗ ∈ U .T; X/. Then the equality L Þ 'Þ = L Þ ' + L Þ .'Þ − '/ holds. We first prove that (4.11)

kL Þ .'Þ − '/k → 0:

194

Jun Kawabe

[10]

To this end, take a finite subset {t1 ; t2 ; : : : ; tn } of T and h ∈ U .T / with 0 ≤ h ≤ 1 satisfying (i) and (ii) of Lemma 4.2. Then, assumption (i) of Theorem 4.1 ensures that lim max k'Þ .ti / − '.ti /k = 0:

(4.12)

Þ

1≤i ≤n

We claim that lim sup k.1 − h/.'Þ − '/k∞ ≤ 2":

(4.13)

Þ

Sn

In fact, let t ∈ i =1 V .ti /. Then, there exists i 0 with 1 ≤ i 0 ≤ n such that .ti 0 ; t/ ∈ V . n Hence, by (4.10) we have supt ∈Si=1 V .ti / k'Þ .t/−'.t/k ≤ 2"+max1≤i ≤n k'Þ .ti /−'.ti /k n for each Þ. Thus, by (4.12) we have lim supÞ supt ∈Si=1 V .ti / k'Þ .t/ − '.t/k ≤ 2". On the other hand, since the function h satisfies (i) of Lemma 4.2, we have k.1 − h/.'Þ − '/k∞ ≤

sup

t∈

Sn

i=1

V .ti /

k'Þ .t/ − '.t/k

for all Þ, which implies (4.13). Now, observe that kL Þ .'Þ − '/k ≤ kL Þ .h.'Þ − '//k + kL Þ ..1 − h/.'Þ − '//k for each Þ. Recalling assumption (ii), we put M = supÞ k'Þ k∞ < ∞. It follows from Proposition 3.2 that

Z

Z

kL Þ .h.'Þ − '//k ≤ k'Þ − 'k∞ · h d¹Þ ≤ .M + k'k∞ / h d¹Þ

: T

T

On the other hand, it follows from [1, Theorem 4] that kL Þ ..1 − h/.'Þ − '//k ≤ k.1 − h/.'Þ − '/k∞ · k¹Þ k.T /: Consequently, for each Þ we have (4.14)

Z

kL Þ .'Þ −'/k ≤ .M +k'k∞ / h d¹ Þ +k.1−h/.'Þ −'/k∞ k¹Þ k.T /:

T

By the assumption of Theorem 4.1, we have limÞ

R T

h d¹Þ =

R T

h d¹ and

lim k¹Þ k.T / = lim k¹Þ .T /k = k¹.T /k = k¹k.T /: Þ

Þ

Thus, it follows from (4.13) and (4.14) that

Z

lim sup kL Þ .'Þ − '/k ≤ .M + k'k∞ / h d¹

+ 2"k¹k.T / Þ T

< .M + k'k∞ /" + 2"k¹k.T /; which establishes (4.11). By Proposition 4.4 and assumptions of Theorem 4.1 we have kL Þ ' − L'k → 0. This together with (4.11) implies kL Þ 'Þ − L'k → 0 and the proof is complete.

[11]


195

5. Weak convergence of injective tensor product measures We first recall the definition of weak convergence of vector measures. Let S be a topological space and X a Banach space. Let {¼Þ } be a net in M .S; X/ and w ¼ ∈ M .S; X/. We say that {¼ R Þ } converges R weakly to ¼, and write ¼Þ −→ ¼, if for each f ∈ C.S/ we have limÞ S f d¼Þ = S f d¼ in the norm of X; see [6, 22]. The following proposition asserts that in the case of positive vector measures, weak convergence follows from the validity of the above convergence for only bounded uniformly continuous functions f on S; see for example [23, Theorem 8.1 (the Portmanteau Theorem)]. PROPOSITION 5.1. Let S be a uniform space with the uniformity U S . Let X be a Banach lattice. Let {¼Þ } be a net in M +.S; X/ and ¼ ∈ M + .S; X/. Assume that ¼ is tight. Then the following two conditions are equivalent: R R .i/ For every f ∈ U .S/, we have R S f d¼Þ → R S f d¼. .ii/ For every f ∈ C.S/, we have S f d¼Þ → S f d¼. PROOF. It is obvious that (ii) implies (i), and hence we prove (i) implies (ii). Fix " > 0 and f ∈ C.S/. Then, the tight measure ¼ satisfies k¼k.S − K / < " for some compact subset K of S, and it follows from [21, Lemma V.1, page 250] that there exists an open, symmetric set V ∈ U S satisfying (5.1)

| f .s/ − f .s 0 /| < "

whenever .s; s 0 / ∈ V and s 0 ∈ K :

Further, there is a function g ∈ U .S/ such that g.s/ = f .s/ for all s ∈ K and kgk∞ = sups∈K | f .s/|; see [4, Chapter IX, Exercises, Section 1, no. 22]. Then, there exists an open, symmetric set V 0 ∈ U S such that |g.s/ − g.s 0 /| < "

(5.2)

whenever .s; s 0 / ∈ V 0 :

Put W = V ∩ V 0 . Then, W is also an open, symmetric subset of U S , and (5.1) and (5.2) remain true for the set W instead of V and V 0 , respectively. Put K .W / = {s ∈ S : .s; s 0 / ∈ W for some s 0 ∈ K }. Then, by similar arguments as in the proof of (4.3) and (4.5) of Proposition 4.3, it follows from (5.1), (5.2), the positivity of ¼ and assumption (i) that (5.3)

sup | f .s/ − g.s/| ≤ 2"

s∈K .W /

and

lim sup k¼Þ k.S − K .W // ≤ ": Þ

Now, for each Þ we have

Z

Z

Z

Z Z Z

(5.4) f d¼

f d¼Þ −

≤ f d¼Þ − g d¼Þ + g d¼Þ − g d¼ S S S S S S

Z

Z

+ g d¼ − f d¼

: S

S

196

Jun Kawabe

By the first inequality of (5.3), for each Þ we have

Z

Z

f d¼Þ − g d¼Þ (5.5)

S S

Z

Z

Z

≤ f d¼Þ + g d¼Þ

+ S−K .W /

S−K .W /

[12]

K .W /

. f − g/ d¼Þ

≤ k f k∞ k¼Þ k.S − K .W //+kgk∞ k¼Þ k.S − K .W // + 2"k¼Þ k.S/: Similarly, noting that k¼k.S − K / < " and g = f on K , for each Þ we have

Z

Z

g d¼ −

≤ " .k f k∞ + kgk∞ / : (5.6) f d¼

S

S

On the other hand, it follows from R assumption R (i) that limÞ k¼Þ k.S/ = limÞ k¼Þ .S/k = k¼.S/k = k¼k.S/

R and limÞR S g d¼ Þ = S g d¼. This together with (5.3)–(5.6) implies lim supÞ S f d¼Þ − S f d¼ ≤ 2" .k f k∞ + kgk∞ + k¼k.S//, which establishes (ii). LEMMA 5.2. Let S and T be uniform spaces with the uniformities U S and UT , respectively. Let X be a Banach space. Assume that a net {¼Þ } in M .S; X/ is Runiformly bounded, that is, supÞ k¼Þ k.S/ < ∞. Let h ∈ U .S × T / and put 'Þ .t/ = h.s; t/¼Þ .ds/ for all Þ and t ∈ T . Then, {'Þ } satisfies assumptions (ii) and (iii) of S Theorem 4.1. PROOF. The uniform boundedness of {'Þ } follows from [2, Theorem 2.6]. Put M = supÞ k¼Þ k.S/ < ∞. Given " > 0, chose V ∈ US and W ∈ UT such that |h.s; t/ − h.s 0 ; t 0 /| < "=.M + 1/ whenever .s; s 0 / ∈ V and .t; t 0 / ∈ W . Thus, if .t; t 0 / ∈ W , then for all Þ we have

Z

0 0

k'Þ .t/ − 'Þ .t /k = .h.s; t/ − h.s; t //¼Þ .ds/

S

≤

" " · k¼Þ k.S/ ≤ · M < "; M +1 M+1

which implies the uniform equicontinuity of {'Þ }. Let S and T be uniform spaces. Let X and Y be Banach lattices. Then, by [8, Theorem], given vector measures ¼ ∈ M .S; X/ and ¹ ∈ M .T; Y /, there exists e ¹ : B .S/ × B .T / → X ⊗ e Y , which is called the a unique vector measure ¼ ⊗ e ¹/.A × B/ = ¼.A/ ⊗ ¹.B/ for all injective tensor product of ¼ and ¹, such that .¼ ⊗ A ∈ B .S/ and B ∈ B .T /; see also [16, Theorem]. In this section, as an application of Theorem 4.1, we consider a problem of joint continuity of the injective tensor product of positive vector measures in certain Banach lattices.

[13]


197

In the rest of this paper, we assume that S and T satisfy B .S × T / = B .S/ × B .T / (it is routine to check that this condition is satisfied, for instance, either S or T has a countable base of open sets). In this case, we can view the injective tensor product e ¹ as a vector measure defined on B .S × T /, and integrate any (uniformly) ¼⊗ e ¹. As an application of Theorem 4.1 we continuous real function with respect to ¼ ⊗ have the following result. THEOREM 5.3. Let S and T be uniform spaces. Let X and Y be Banach lattices. Let {¼Þ } be a net in M +.S; X/ and ¼ ∈ M +.S; X/. Let R{¹Þ } be a net inR Mt+ .T; Y / + Rand ¹ ∈ MRt .T; Y /. Assume that ¹ is − -smooth. If S f d¼ R Þ → S f d¼ and e RT g d¹Þ → T g d¹ for every f ∈ U .S/ and g ∈ U .T /, then S×T hd.¼Þ ⊗ ¹Þ / → e ¹/ for every h ∈ U .S × T /. hd.¼ ⊗ S×T PROOF. We may assume without loss of generality that {¼Þ } and {¹Þ } are uniformly bounded, that is, supÞ k¼Þ k.S/ < ∞ andR supÞ k¹Þ k.T / < ∞. R Fix h ∈ U .S × T /, and put 'Þ .t/ = S h.s; t/¼Þ .ds/ and '.t/ = S h.s; t/¼.ds/ for each Þ and t ∈ T . Then, by Lemma 5.2 and the assumptions of this R theorem, {'Þ } e d¹Þ → and ' satisfy the conditions of Theorem 4.1, and hence we have ' ⊗ T Þ R e ' ⊗ d¹. By the Fubini theorem for injective tensor product measures (see, T R R e e for R instance, R[11, Theorem 13]), we have T 'Þ ⊗ d¹Þ = S×T hd.¼Þ ⊗ ¹Þ / and e d¹ = S×T hd.¼ ⊗ e ¹/, which proves the theorem. '⊗ T eY Let X and Y be Banach lattices. Then, in general, the injective tensor product X ⊗ or the projective tensor product may not be a vector lattice for the natural ordering. However, the injective tensor products of some important examples of Banach lattices are also Banach lattices; see the examples after the proof of the following theorem. e Y is also Let X and Y be Banach lattices such that the injective tensor product X ⊗ a Banach lattice satisfying the condition x ⊗ y ≥ 0 for every x ≥ 0 and y ≥ 0. Let .; A / and .0; B / be measurable spaces. Let ¼ : A → X and ¹ : B → Y be vector measures. Then it is easy to verify that if ¼ and ¹ are positive, so is the injective e ¹. In this case, we have an affirmative answer for a problem of tensor product ¼ ⊗ joint continuity of the injective tensor products with respect to the weak convergence of vector measures. THEOREM 5.4. Let S and T be uniform spaces. Let X and Y be Banach lattices e Y is also a Banach lattice satisfying the such that the injective tensor product X ⊗ condition x ⊗ y ≥ 0 for every x ≥ 0 and y ≥ 0. Let {¼Þ } be a net in M + .S; X/ and ¼ ∈ Mt+.S; X/. Let {¹Þ } be a net in Mt+ .T; Y / and ¹ ∈ Mt+.T; Y /. Assume that ¹ w w w e ¹Þ −→ e ¹. is − -smooth. If ¼Þ −→ ¼ and ¹Þ −→ ¹, then ¼Þ ⊗ ¼⊗ R e R PROOF. By Theorem 5.3, for every h ∈ U .S × T / we have S×T hd.¼Þ ⊗ ¹Þ / → e ¹Þ ’s and ¼ ⊗ e ¹/. By assumption, ¼Þ ⊗ e ¹ are positive, and it is easy d.¼ ⊗ S×T

198

Jun Kawabe

[14]

e ¹ is tight. Consequently, it follows from Proposition 5.1 that to prove that ¼ ⊗ w e ¹Þ −→ e ¹. ¼Þ ⊗ ¼⊗ REMARK. In the special case that X = Y = R, an alternative proof of Theorem 5.4 is executed by a well-known criterion that one can prove the weak convergence of ¼Þ to ¼ by showing that ¼Þ .A/ → ¼.A/ for some special class of sets A (see, for instance, [24, Corollary 1 to Theorem I.3.5 and Proposition I.4.1]). However, it seems that the usual proof of the above criterion does not work well for positive vector measures, since the notions of limit infimum and limit supremum cannot be extended to general Banach lattices. We finish this paper with examples of Banach lattices X and Y such that the injective e Y is also a Banach lattice satisfying the condition x ⊗ y ≥ 0 for tensor product X ⊗ every x ≥ 0 and y ≥ 0; see examples in [20, pages 274–276] and [12, page 90]. eY EXAMPLES. (1) If K is a compact space and Y be any Banach lattice, then C.K / ⊗ is isometrically lattice isomorphic to the Banach lattice C.K ; Y /. Especially, when e C.L/ is isometrically lattice isomorY = C.L/ for some compact space L, C.K / ⊗ phic to C.K × L/. (2) Let P be a locally compact space and Y be any Banach lattice. Denote by C0 .P; Y / the Banach lattice with its canonical ordering of all continuous functions ' : P → Y such that for every " > 0, the set {s ∈ P : k'.s/k ≥ "} is compact. e Y is isometrically lattice isomorphic We write C0 .P/ = C0 .P; R/. Then C0 .P/ ⊗ to C0 .P; Y /. Especially, when Y = C 0 .Q/ for some locally compact space Q, e C0 .Q/ is isometrically lattice isomorphic to C0 .P × Q/. C0 .P/ ⊗ (3) Let .; A ; ¦ / be a measure space and Y be any Banach lattice. Denote by L ∞ .; Y / the Banach lattice of all (equivalence classes of) ¦ -essentially bounded measurable functions ' : → Y with its canonical ordering. We write L ∞ ./ = e Y is a Banach lattice. However, in general, L ∞ ./ ⊗ eY L ∞ .; R/. Then, L ∞ ./ ⊗ ∞ is a proper closed subset of L .; Y /. Acknowledgement The author is very grateful to the referee for detailed comments and suggestions which have significantly improved this paper. References [1] R. G. Bartle, ‘A general bilinear vector integral’, Studia Math. 15 (1956), 337–352. [2] R. G. Bartle, N. Dunford and J. Schwartz, ‘Weak compactness and vector measures’, Canad. J. Math. 7 (1955), 289–305.

[15]


199

[3] P. Billingsley, Convergence of probability measures (Wiley, New York, 1968). [4] N. Bourbaki, Topologie ge´ ne´ rale, 2nd edition (Hermann, Paris, 1958). [5] I. Csisza´ r, ‘On the weak∗ continuity of convolution in a convolution algebra over an arbitrary topological group’, Studia Sci. Math. Hungar. 6 (1971), 27–40. [6] M. Dekiert, Kompaktheit, Fortsetzbarkeit und Konvergenz von Vectormaßen (Dissertation, University of Essen, 1991). [7] J. Diestel and J. J. Uhl, Jr., Vector measures, Amer. Math. Soc. Surveys 15 (Amer. Math. Soc., Providence, 1977). ˇ [8] M. Duchonˇ and I. Kluva´ nek, ‘Inductive tensor product of vector-valued measures’, Mat. Casopis Sloven. Akad. Vied. 17 (1967), 108–112. [9] R. M. Dudley, Real analysis and probability (Wadsworth & Brooks/Cole, California, 1989). [10] N. Dunford and J. T. Schwartz, Linear operators. Part 1: General theory (Wiley, New York, 1958). [11] F. J. Freniche and J. C. Garc´ıa-Va´ zquez, ‘The Bartle bilinear integration and Carleman operators’, J. Math. Anal. Appl. 240 (1999), 324–339. [12] A. Grothendieck, Produits tensoriels topologiques et espaces nucle´ aires, Mem. Amer. Math. Soc. 16 (Amer. Math. Soc., Providence, 1955). [13] I. M. James, Topological and uniform spaces (Springer, New York, 1987). [14] J. Kawabe, ‘Weak convergence of tensor products of vector measures with values in nuclear spaces’, Bull. Austral. Math. Soc. 59 (1999), 449–458. , ‘A type of Strassen’s theorem for positive vector measures with values in dual spaces’, [15] Proc. Amer. Math. Soc. 128 (2000), 3291–3300. [16] I. Kluva´ nek, ‘On the product of vector measures’, J. Austral. Math. Soc. 15 (1973), 22–26. [17] I. Kluva´ nek and G. Knowles, Vector measures and control systems (North Holland, Amsterdam, 1975). [18] D. R. Lewis, ‘Integration with respect to vector measures’, Pacific J. Math. 33 (1970), 157–165. [19] M. Ma¨ rz and R. M. Shortt, ‘Weak convergence of vector measures’, Publ. Math. Debrecen 45 (1994), 71–92. [20] H. H. Schaefer, Banach lattices and positive operators (Springer, Berlin, 1974). [21] L. Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures (Tata Institute of Fundamental Research, Oxford University Press, 1973). [22] R. M. Shortt, ‘Strassen’s theorem for vector measures’, Proc. Amer. Math. Soc. 122 (1994), 811– 820. [23] F. Topsøe, Topology and measure, Lecture Notes in Math. 133 (Springer, New York, 1970). [24] N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, Probability distributions on Banach spaces (D. Reidel Publishing Company, 1987). [25] M. Va¨ th, Volterra and integral equations of vector functions (Marcel Dekker, New York, 2000). [26] A. J. White, ‘Convolution of vector measures’, Proc. Roy. Soc. Edinburgh Sect. A 73 (1974/75), 117–135.

Department of Mathematics Faculty of Engineering Shinshu University 4-17-1 Wakasato Nagano 380-8553 Japan e-mail: [email protected]