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For positive operators between Banach lattices a concept of absolute continuity is considered. On an AM-space with order unit norm, S is shown to be absolutely continuous with respect to T if and only if there is an approximation of S by finite sums of operators of the type q o T o h where h and q are multiplication operators or orthomorphisms. Given T compact, compactness of S is characterized. For positive operators S and T from a Dedekind complete Banach lattice E to a Banach lattice F, a concept of absolute continuity is considered. It is shown that for AM-spaces with order unit norm, S is "r-absolutely continuous" with respect to T if and only if S can be approximated by finite sums of operators of the type q o T o h where h and q are orthomorphisms on E and F respectively. The symbol "o" denotes composition so that for f E E, (q o T o h)(f) = q(T(h(f))). In particular E will be identified with functions on a topological space X and h will be identified with a multiplication by a function on X, again denoted by h, so that (hf )(x) = h(x) f (x) for x E X, and similarly for q. The operators r-absolutely continuous with respect to T contain the operators less than or equal to a multiple of T. Theorem 3 relating compactness of S to that of T might be contrasted to a theorem of Dodds and Fremlin (see [3 or 4]) which states, in the presence of order continuous norms in E' and F, that the compactness of T implies the compactness of S 0 in E. This was studied in the setting of Dedekind complete Riesz spaces. We will consider the following analogue in the setting of Banach lattices. We define S to be topologically absolutely continuous with respect to T (or rabsolutely continuous) if for each positive element f in E the element of Sf is in the closure of the order ideal generated by Tf. The condition S less than or equal to a multiple of T implies r-absolutely continuous which in turn implies absolutely continuous (in the sense of Luxemburg). Let C(X) denote the Banach lattice of all continuous real-valued functions on a compact Hausdorff topological space X endowed with the supremum norm (11 lloo) For linear functionals on C(X), given X extremally disconnected (equivalently, C(X) Dedekind complete), both concepts of absolute continuity coincide with the usual notion of absolute continuity for measures. The extremally disconnected assumption will be needed for the proof of the following proposition and in turn for Received by the editors February 10, 1986 and, in revised form, May 5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B30; Secondary 47B05, 54C40. ?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

Operators on Banach lattices and the Radon-Nikodým theorem