#TITLE_ALTERNATIVE# - DARMAWIJAYA, SOEPARNA
For every n let the function fn be integrable in some sense on [a,b] and the sequence {fn} convergent to f almost everywhere on [a,b]. A common but interesting problem is to find sufficient conditions in order that f will be integrable on [a,b] and $bf(t)dt = 1 i m abfn(t)dt. In the n + 0 case of Lebesque's integral, several such sets of conditions have been established, for instance that all the functions f are dominated by one integraible function or, that all n the primitives of fn are uniformly absolutely continuous on [a,b]. In the case of Henstock's integral, it is sufficient that the sequence {fn} converges in the controlled sense or, that all the functions are dominated by some integral function. This dissertation is concerned with seeking solutions for the above problem in case of the approximately continuous integral of Bullen. This integral is based on the concept of an approximately full cover, which is an approximate generalization of a full cover. Therefore Bu11en's approximate continuous integral is the approximate generalization of Henstock's integral. It is therefore logical that the search for those sufficient conditions starts with formulating approximate generalizations for concepts and notions assosiated with the Henstock's integral and convergence of functions. The new concepts include : limits, derivatives, primitives, continuity, absolute continuity, local convergence and controlled convergence of functions. Let Rap[a,b] denote the class of all approximately continuously integrable functions on [a,b] and for / E Rap[a,b] let CRap)abf(t)dt be its Bullen integral over [a,b]. The main result of this dissertation can be formulated as follows. Suppose the sequence of functions {fn} in R* [a,b] converges almost everywhere on [a,b] to the function f and for each n let Fn be its Bullen Rap-primitive. Then, the three following conditions are equivalent and are sufficient to insure that f e R* [a,b] and Based on the above main results, we may construct a type Riesz definition of the approximately continuous integral of Bullen as follows. A function f is said to be RDap-integrable on[a,b] if there exists a sequence of simple functions On which is approximately controlled convergento f on [a,b], i.e., {cn} converges to f almost everywhere on [a,b] and satisfies the condition (a). Using RD*-integral, the Riesz type definition of the approximately continuous integral of Bullen can be formalated as follows. A function f is approximately continuous integrable of Bullen on [a,b] if and only if f is RD*-integrable on [a,b].