COVERING SYSTEMS AND DERIVATES IN HENSTOCK DIVISION SPACES B. S. THOMSON The theory of division spaces introduced by Henstock in [2, 3] in order to simplify and unify certain areas in the theory of integration is particularly well suited to accomodating various parts of the theory of covering systems and differentiation. In this paper we present an introduction to these ideas in a quite general setting. Our approach involves the elegant idea due to Henstock in [1] of introducing a further measure on the space, called the inner variation; proving various results on derivates which hold almost everywhere with respect to this measure; and then imposing " Vitali" conditions to assure that the measure coincides with the original one. 1. Division spaces, variation Let T be a set and I a collection of pairs (/, x) (I ^ T, x e T). A finite subset D of I is said to be a division if the sets in {/ : (/, x) e D} are disjoint. For a division D we write <r(D) = \J{I: (/, x)e D} and we call any set E = a(D) an elementary set and D a division of E. c T and S s I we define (1.1) S(*) = {(/,*) eS: /£*}, (1.2) S[X] = {(J,x) e S: * e *}- Definition 1. The ordered triple (T, % I) is said to be a division system provided $1 is a family of subsets of I such that (i) If x e T and S e 9t then (0, JC) e S. (ii) 91 is directed downwards by set inclusion. Condition (i) is unnecessary but makes the covering system definitions of the next section easier to apply. A division system (T, 5t,1) is said to be fully decomposable (resp. decomposable) if to every family (resp. countable family) {X t : i e/} of disjoint subsets of T and every {S t : iel} s $t there exists an SeSI with S[Z,] s S,[X,] for all iel. If fi is a real-valued function defined on I we define the variation of n with respect to an S £ I as (1.3) F(^S) = sup(D)HM/,*)l, where the supremum is taken over all divisions D ( D g S ) and (D) X denotes summation over all (/, x) e D, an empty sum by convention being zero. If $( is a family of subsets of I then we define also (1.4) V0x,2l) = inf{70i,S): Se$l}. Received 6 April, 1970; revised 22 October, 1970. [J. LONDON MATH. SOC. (2), 4 (1971), 103-108]