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However, the Bochner Integral Is Quite Re- Strictive REPRESENTATIONS OF WEAK AND STRONG INTEGRALS IN BANACH SPACES By JAMES K. BROOKS DEPARTMENT OF MATHEMATICS, UNIVERSITY OF FLORIDA Communicated by E. J. McShane, February ,4, 1969 Abstract.-We establish a representation of the Gelfand-Pettis (weak) in- tegral in terms of unconditionally convergent series. Moreover, absolute con- vergence of the series is a necessary and sufficient condition in order that the weak integral coincide with the Bochner integral. Two applications of the representation are given. The first is a simplified proof of the countable addi- tivity and absolute continuity of the indefinite weak integral. The second ap- plication is to probability theory; we characterize the conditional expectation of a weakly integrable function. 1. Introduction.-In this paper we give a representation theorem for measur- able Gelfand-Pettis integrable functions in terms of unconditionally convergent series. As we shall explain in more detail below, this representation is optimal. The advantage of the Gelfand-Pettis (G-P) integral is that convergence in the Banach space is transferred to convergence in the scalar field. The disadvantage in working with this integral was the lack of a useful representation (e.g., see ref. 11, p. 77); consequently, it has not been widely used in applications, al- though it arises in a very natural way. As a result, the Bochner integral2-or strong integral-is generally used; however, the Bochner integral is quite re- strictive. The representation theorem (Theorem 1) now facilitates the use of the G-P integral. In fact, we shall show that the usual proofs9. 4 Of the deep theorem that the G-P integral is absolutely continuous and countably additive can be considerably simplified now. The vector integration theories of BirkhoffI and Dunford (the second integral and the absolutely continuous third integrals) coincide with the G-P theory in the case of measurable functions."4 Consequently, our result applies to these integrals. The conditional expectation of a G-P integrable function is presented in section 3. Theorem 2 suggests an approach to integration in linear topological spaces. It will be shown elsewhere that our representation theorem is equiva- lent to the Orlicz-Pettis theorem.6 We now fix some notation. X is a Banach space over the complex numbers with conjugate space X*. (S, 2, IA) is a a-finite extended real valued measure space; Iu denotes the total variation function of 1A. Without loss of generality, we assume that the measure space is complete. f: S -X I is measurable if it is the a.e. IA limit of a sequence of simple functions. tE denotes the characteristic func- tion of E. f: S -X I is Gelfand-Pettis integrable10. 14 or weakly integrable (with re- spect to ;u) is: (1) x*f is is-integrable for every x* E X*; (2) for every E E 2, there exists an element XE in X such that x*(xB) = fE x*f du, for every x* E I. In this case, x, is defined to be the weak integral of f over E; in symbols: XE = (X) fE f dis. If f is Bochner integrable, the symbol (X) is omitted. 266 Downloaded by guest on September 29, 2021 VOL. 63, 1969 MATHEMA TICS: J. K. BROOKS 267 2. THEOREM 1. Let f:S -* X be a measurable weakly integrable function. Then f can be represented in the form f = g + h a.e. gu, where g is a bounded Bochner integrable function and h assumes at most countably many values in X. If one writes h in the form h = 2 x1x , where the measurable sets E1 are disjoint, then (X)fREfdsl =fHe9g d$+ Exi(Ei n E), (#) where the last series converges unconditionally for each E C 2. It converges abso- lutely if and only if f is Bochner integrable. Remark: Conversely, if f can be represented in the above form, and the series in (#) converges weakly to an element in X for each E E 2, then f is weakly integrable and measurable. Proof: We first show that a measurable function is equal a.e. 1i to the sum of a bounded measurable function, with a prescribed positive bound, and a func- tion that has at most a countable range. Since f is the a.e. 1A limit of a sequence of simple functions, we can choose a function which equals f a.e. ,u and has a separable range. For notational convenience, assume that f has this property and that I,.l (S) < co. Let { an} be a decreasing sequence of positive numbers such that 2' i an < a). Let S(n, f(s)) denote the open sphere of radius an with center at f(s). Since f(S) C U88s S(n, f(s)), by the Lindelkf theorem for each n there exists a sequence { s'}'.i 1 C S such thatf(S) C U . = 1 S(n,f(s?)). Since the function f - f(s) is a real valued measurable function, If - f(Se')JI-1 [0, an) = A' E 2. Let E,, =A -((U 1 An) . Definefn = 2 = 1 f(s )tE Since If(s) - fn(s)jf < an on S, fn -O-f uniformly on S. Define g(s) = 2= 2 (n(S) -fn-(s)). Note that Ilg(8)I < Yn = 2IIfn(5) -fn-i(s)I . 2 n= 1 an 5; also,fsjlg11 dIu| < (2 2 = lan)IMA (S). The proof for the a--finite case will be omitted. Hence, g is a bounded Bochner integrable function. Let xi f(s4) and let Et = El. Define h = 7= 1xitE. Note that the E2 are disjoint; also,f = g + h. Sincef and g are weakly integrable, h is weakly integrable; consequently, for every E C 2, there is a ZB CE I such that X*(ZR) = fEx*h dM&, x* C X*. Thus, ZIx*(xJ)!I MIA(Ei n E) = fEIx*hI dIMIA < o. (1) We now show that the series Zx#(E1 n E) (2) i = 1 converges unconditionally in X for every E. By the Pettis-Orlicz theorem,6 it suffices to show that every subseries of (2) converges weakly to an element in X. Let 7r denote a subsequence of the natural numbers. Define A = UiE r Ei. Then x*(zEnA) = fEnAX*h dM = 2 x*(xi)A(E n A n Ei) = 2i, x*(xi) ,g(E n Et), which converges, because of (1), for each x* E if*. Hence, (2) con- verges unconditionally, and (#) holds. The fact that a measurable weakly integrable function is Bochner integrable if and only if 2|IX|I| IM(E n E,)| < X, E E Z, is deduced as follows. Iff is Boch- ner integrable, then ZiIxixl l4Eln Et)| < Z4Ixill IMIA(Ei) = fsbf -gI|dI|I < co. Suppose 2lIx{IlI I1(E n E,) < a,,E CE . LetthemeasurablesetsA andB form Downloaded by guest on September 29, 2021 268 MATHEMATICS: J. K. BROOKS PROC. N. A. S. a Hahn decomposition of (S, 2, JA); A and B are positive and negative sets, re- spectively. ThenfsfIhfl dIuI = 2;4f|x4j ftj(Ei) = 2z|fx4J|,i(E, n A) -2M||x| I (E, n B) < c. Hence, h is Bochner integrable, and the conclusion follows. Remarks: (1) The above representation is optimal in the following sense. The structure of the bounded Bochner integrable function g is fairly well known. The remainder, f - g = h, could not, in general, have a finite range, for other- wise the weak and strong integrals would always coincide, which they do not in general.' It follows from Theorem 2 below that weak integrals can be described in terms of unconditionally convergent series. The difference between the classes of weakly integrable functions and strongly integrable functions is the class of infinite series in a Banach space that converge unconditionally but not absolutely. (2) It follows from the proof of the above theorem that every measurable func- tion is equal a.e. ;A to a bounded measurable function and a function which has at most a countable range; the bound on g can be made arbitrarily small. By using this decomposition, one can avoid the method of truncation in defining the integral of a summable function. The definition of the integral of a bounded measurable function is relatively simple, and the integral of the remainder can immediately be written, viz., Zixi(E1); hence, the definition of the integral is technically easier when the decomposition is used. THEOREM 2. Let f: S -X T be measurable. f is weakly integrable if and only if: (1) f is expressible in the form (*) f = 2; = iy4tA1 a.e. IAy iA1e 2AE ,wheretheseries converges absolutely a.e. ,t; (2) 2;, 5y j(E n Ai) converges unconditionally for every E E 2;. In this case, (X)fjfd; = 2;T=yij(E n A1). f is Bochner integrable if and only if there is a representation (*) for f such that 2= I||yi| A (A j) < co Proof: The proof follows from Theorem 1 and its proof, the Beppo Levi theorem, and the representation of a Bochner integrable function in terms of an infinite series. Remarks: (1) It is now an easy matter to give a simple definition of a weakly integrable function-and its integral-in the case S is Euclidean n-space, 2; is the class of Lebesgue measurable subsets of R , and ; is n-dimensional Lebesgue measure. The method avoids the use of measure and continuous linear func- tionals by the use of J. Mikusin'ski's elegant definition of the Bochner integral on Rn.12, 13 Details will be presented elsewhere. (2) The simple form of (*) and of its integral can be used in transforming vector integrals defined on Rn (cf.
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