espect to the X-valued measure Received by the editors August 1995. Communicated by J. Schmets. 1991 Mathematics Subject Classification : Primary 46G10, 47B40; Secondary 46A08. Bull. Belg. Math. Soc. 3 (1996), 267--279 Px (for each x X) than to establish integrability directly with respect to the L s (X)-valued measure P. The most general result concerned with this question (for f satisfying f X) is Proposition 1.2 of [5], which states that P [f ] given by (1) is continuous whenever the following conditions are satisfied: (2.i) X is quasicomplete, (2.ii) L s (X) is sequentially complete, and (2.iii) P(#) = (E); E #} is an equicontinuous subset of L(X). The proof of this result is based on an elegant automatic continuity result of P.G. Dodds and B. de Pagter, [3; Corollary 5.7], which states (under (2.i) and (2.ii)) that if M is a strongly equicontinuous Boolean algebra of projections (M = P (#) in (2.iii) satisfies this) in L(X), then an everywhere defined linea