INTEGRAL REPRESENTATION OF LINEAR CONTINUOUS OPERATORS FROM THE SPACE OF LEBESGUE-BOCHNER SUMMABLE FUNCTIONS INTO ANY BANACH SPACE* BY WITOLD M. BOGDANOWICZ

CATHOLIC UNIVERSITY OF AMERICA, WASHINGTON, D. C. Communicated by A. Zygmund, June 9, 1965 Let R be the space of reals and Y, Z, W be Banach spaces. Denote by U the space of all bilinear continuous operators u from the space Y X Z into W. Norms of elements in the spaces Y, Z, W, U will be denoted by I . A family of sets V of an abstract space X is called a pre-ring if the following conditions are satisfied: (a) if A1,A2 C V, then A, n A2 C V; (b) if A1,A2 C V, then there exist disjoint sets B1, ..., Bk C V such that A1\A2 = B1U ... U Bk. A Au from a pre-ring V into a Banach space Z is called if it satisfies the following conditions: for every countable family of disjoint sets A, E V (t E T) such that (c) A = U A, C V, we have 1u(A) = E ,u(A ,) where the last sum is con- T T vergent absolutely and ,(A) = sup{ |jM(A,) } < o for any A C V, where the supremum is taken over all possible decompositions of the into the form(c). A volume is called positive if it takes on only nonnegative values. If 1A is a volume, then 1 is a positive volume. Let v be a positive volume defined on the pre-ring V. Denote by M the space of all 1A from the pre-ring V into the space Z, such that 1,4u(A)I < cv(A) for all A C V and some c. The least constant c satisfying the inequality is denoted by |i/l . It is easy to see that (M, llj1|l) is a Banach space. In reference 1 an integral f u(f,d,4) has been defined from the space U X L(Y) X M into the space W, where L(Y) denotes the space of all functions from X into Y summable with respect to v.2 The norm on the space L(Y) is defined by the formula If1 = fl f(*)Idv. The integral represents a trilinear operator and it has the property: Ifu(fd,4)I < Iu| f|fl,||t|1 for u E UfC L(Y),s E M. Abbreviations of the form Th. 1, Th. 3, L. 1, and so on will refer to results of reference 1. Let Z denote the set of all linear continuous operators z from the space Y into the space W. Let uo E U be a fixed bilinear form defined by the formula uo(y,z) = z(y) for y C Y, z C Z. Denote by H the space of all linear continuous operators h from the space L(Y) into the space W. THEOREM 1. To every linear continuous operator h E H corresponds a unique vector-valued volume ,u C M such that h(f) = fuofdi) for all f E L(Y). 351 Downloaded by guest on September 29, 2021 352 MATHEMATICS: W. M. BOGDANOWICZ PROC. N. A. S.

This correspondence establishes an isometry and an isomorphism between the space H and the space M. Proof: Take any A E V, y F Y. We have XAY E L(Y). Let ZA(Y) = h(xAyY) We see that ZA(y) < IIh Iv(A)I yI for all y F Y. Therefore, ZA F Z. Let A = U Aj E V, where Aj F V (j = 1, ..., n) is a finite family of disjoint sets. Since XAY = XA1Y + . . . + XAny and the functions XAY, XAiY are in the space L(Y), therefore from the additivity of the operator h we get E>jZAj(Y) = ZA(Y) for all y E Y. Define a function , from the pre-ring V into the space Z by the formula ,u(A) = ZA for A E V. From previous results we see that the function ,u is additive. LEMMA. Let jA be a function from the pre-ring V into the space Z. If ,A(A) is finite-additive and I#u(A)I < cv(A) for all A E V, then ,A E M and flufl < c. Proof: Take any decomposition A = U An E V into a countable family of disjoint sets An E V. It follows from the definition of a pre-ring that for every positive integer m there exist disjoint sets Ci E V (i = 1, .I, k) such that A \(A1U .. UAm) = C1 U ... U Ck. It follows from finite-additivity that 8(A) = /1(Ai) + ... + M(Am) + A(C1) + ... + A(Ck) and v(A) = v(Al) + ... + v(Am) + v(Cl) + + v(Ck). From sigma-additivity of the volume v we have v(A) = v(Al) + ... + v(Am) + v(Am+i) + Therefore, I(A) - (U4(A1) + ... + g(Am))| < cEn>mv(An). But this proves sigma-additivity of the functionu. We easily notice that ,u (A) < c v(A), and therefore EF JUT and fl '4 < c Thus, the lemma is proved. Now take any A F V and y C Y, IIyI < 1. We have jA(A) (y) = h(XAY) .< h| v(A) for A C V, and therefore ,(A)I <.Ih|| v(A) for all A F V. According to the lemma this implies (i) u lFC I and I|Ihu< ||h||. Take any simple function s = XAY1 + . . . + XAnYn, AI F V, y, F Y. We have

h(s) = h(XAiYl) + ... + h(XAJy.) = IA(Ai)(yi) + ..- + ,A(An)(Y.) = fuo(s,dIA). Downloaded by guest on September 29, 2021 VOL. 54, 1965 MATHEMATICS: W. M. BOGDANOWICZ 353

Now if f E L(Y), then by the definition of the space L(Y) (see ref. 1) there exists a of simple functions so such that IIsn - fj 0 (L. 4). Now from Th. 1 and continuity of h we have h(f) = lim h(sn) = lim fuo(sn,dA) = uo(f,dM). Therefore, h(f) = f uo(fdA) for all f F L(Y). To prove the uniqueness of the representation, take any volume ,u F M. If h(f) = fuo(f,d~j) for all f E L(Y), then f uo(f,d(M - ii)) = 0 for all f E L(Y). Sincef = XAY F L(Y) for all A E V and y E Y, therefore ,u(A) (y) - A(A) (y) = UO(XAYd(,4 -8)) = 0. Thus, A = A. Now from the representation h(f) = fuo(fdt) and Th. 1, we have Ih(f)I = Ifuo(fdA)I .< uol fil 11KI1- Since uo = 1, therefore h ,I | . The last inequality together with the in- equality (i) yields jh| = 11,u1 |. We have proved that to every linear continuous operator h there corresponds a unique volume /u, and conversely. This correspondence is linear and preserves the norm. This concludes the proof of the theorem. THEOREM 2. Let A,u,, F A1 be the volumes corresponding to the operators h,h, E H. Then hn(f) converges to h(f) for all f E L(Y) if and only if the sequence An is bounded and A,,(A)(y) converges to ,A(A)(y) for every A F V and y E Y. Proof: The necessity of the condition is obvious. To prove that it is sufficient, it is enough to notice that the linear combinations of the functions of the form XAY, A F V, y C Y, form a dense set in the space L(Y). The proof is similar to the proof of Th. 6.

* This work was partially supported by NSF grant GP 2565. 1 Bogdanowicz, W. M., "A generalization of the Lebesgue-Bochner-Stieltjes integral and a new approach to the theory of integration," these PROCEEDINGS, 53, 492-498 (1965). 2 Also cf. Bourbaki, N., "Integration," chap. 6, Actualites Sci. Ind. 1281 48-50 (1959). Other sources are: Bogdanowicz, W. M., "Integral representation of linear continuous operators on LP spaces of Lebesgue-Bochner summable functions," to appear in Bull. Acad. Polon. Sci. Bochner, S., and A. E. Taylor, "Linear functionals on certain spaces of abstractly-valued functions," Ann. Math., 39, 913-944 (1938). Dunford, N., "Integration and linear operations," Trans. Amer. Math. Soc., 40, 474-494 (1936). Dunford, N., and B. J. Pettis, "Linear operators on suimmable functions," Trans. Amer. Math. Soc., 47, 323-392 (1940). Fullerton, R. E., "The representation of linear operators from LP to L," Proc. Amer. Math. Soc., 5, 689-696 (1954). Gelfand, I. M., "Abstrackte Funktionen utnd lineare Operatoren," Mat. Sb., 4 (46), 235-286 (1938). Kantorovitch, L. V., and B. Z. V-ulich, "Stir la representation des operations lindaires," Com- positio Math., 5, 119-165 (1938). Phillips, R. S., "On linear transformations," Trans. Amer. Math. Soc., 2, 516-541 (1940). Downloaded by guest on September 29, 2021 354 MATHEMATICS: L. LECAM PROC. N. A. S.

Phillips, R. S., "On weakly compact subsets of a Banach space," Amer. J. Math., 65, 108-136 (1943). Sirvint, G., "Sur les transformations integrales de l'espace L," Dokl. Akad. Nauk SSSR, 18, 255-257 (1938). Yosida, K., Y. Mimura, and S. Kakutani, "Integral operator with bounded kernel," Proc. Imper. Acad. Tokyo, 14, 359-362 (1938).

A REMARK ON THE CENTRAL LIMIT THEOREM* BY LuCIEN LECAM UNIVERSITY OF CALIFORNIA, BERKELEY Communicated by J. Neyman, May 6, 1966 1. Introduction.-If jAl and g2 are two probability measures on the real line, let P(;l,/A2) be the Kolmogorov vertical distance P(/.L1,M2) = SUP lIys (- ,X]} -92{ (-a ,X]}1. x Let {X1; j = 1,2,... .I be a finite set of independent random variables. Let Pj be the distribution of X1 and let P = HPj be the convolution product which is the distribution of 2:X>. Several of the usual central limit theorems can be considered elaborations of the fact that, under suitable assumptions, the distance p(P,Q) be- tween P and the convolution exponential Q = exp { 2(Pj - I)I is small. The approximation theorems given in references 1, 2, and 3, for instance, suggest that p(PQ) will be small if and only if each one of the variables X; is small com- pared to the dispersion of the sum 2X>. Let Rj = HkIq Pk and let r(P,T) be the modulus of continuity r(P,'r) = sup5 P{ (x,x + r]}. Finally, let aj(0) = Pi{ jX|j > 0} and let a(0) = sup} aj(0). The purpose of the present note is to show that when the variables Xj have sym- metric distributions, if any one of the quantities 52 = p(PQ), e = supj p(Rj,P), and info[F(P,0) + a(@) I is small, so are the other two. In particular, p(P,Q) is small when and only when each of the P1 contributes little to the product P. 2. Inequalities for Characteristic Functions.-Consider a finite set {Xi; j = 1,2... I} of independent random variables as above. If ,u is an arbitrary signed meas- ure, let )i be its Fourier transform defined by ,a(t) = feixi(dx). Further, for 0> 0 let Di2(0) be the censored variances defined by DJ2(0)= fmin [1, J2 Pj (dx). For 0 = 0 the value of D12(0) will be taken equal to DO2(0) = aj(O) where aj(0) = PAIXj1 > 01 for 0 > 0. Furthermore, we shall denote B2(0) the sum B2(0) =2j D12(0). Assuming now that the P1 are symmetric around zero, note that if 5a1(0) . 1 Downloaded by guest on September 29, 2021