A SHORT PROOF OF THE CONTROLLED CONVERGENCE THEOREM FOR HENSTOCK INTEGRALS LEE PENG YEE AND CHEW TUAN SENG Recently, Lee and Chew [5] proved a better convergence theorem for Henstock integrals. The proof is rather involved. In this note, we give a short proof of the theorem. We refer to [5] for definitions and notations. A sequence {/„} of real-valued functions is said to be control-convergent t o / o n [a, b] if the following conditions are satisfied: (i) f n (x) ->f{x) almost everywhere in [a, b] as n -> oo where each/ n is Henstock integrable on [a, b]; (ii) the primitives F n of/„ are ACG, uniformly in n, that is, [a, b] is the union of a sequence of closed sets X t such that on each X { the functions F n are AC»(Jj) uniformly in n; (iii) the primitives F n converge uniformly on [a, b]. LEMMA. If a sequence of functions {/„} is control-convergent tofon [a, b], then for each i and for every e > 0 there is an integer N such that for every partial division of [a, b] given by a ^a l < b l ^ a 2 < b 2 ^ ... ^ a p < b p ^: b with a,, b x ,a 2 ,b 2 ,...,a p ,b p belonging to X t , we have v £ co( F n — F m ; [a t , b t ]) < e whenever n,m^ N t-i where co denotes the oscillation of F n — F m over [a it b t ]. Proof Since X t is closed, (a, b) — X t is open. We may assume a,beX t and let F(x) be the limit of the sequence F n (x) as n-*co. Then define H n (x) = F n (x) — F(x) when xeX t and piecewise linear on the complement of X t as follows. For every com- ponent interval (c,d) of (a,b) — X t with midpoint e we define H n (e) such that H n {e) > F n (c)-F(c), H n (e) > F n (d)-F(d), and max{H n (e)-F n (c) + F(c), H n (e)-F n (d) + F(d)} = co(F n -F; [c,d]). That is, the oscillation of H n over [c,d] is equal to that of F n — FOVQT [c,d]. Elsewhere H n is defined linearly. Note that the limit of the sequence H n (x) exists as n -> oo for each x and in view of (iii) it is zero. Also, H' n (x) converges almost everywhere as n -* oo. Hence it follows from the proof of Lemma 4 in [5] that the result of the lemma Received 28 February 1986; revised 2 July 1986. 1980 Mathematics Subject Classification 26A39. Bull. London Math. Soc. 19 (1987) 60-62