SOME PROBLEMS IN INTEGRATION THEORY Lee Peng-Yee (received 11 June, 1971) I shall state in what follows some problems in integration theory. They will, I hope, be of interest to the reader. Their solutions, if any, will certainly lead to further development and useful applica­ tions. The integrals involved are the Denjoy, Henstock's Riemann- complete, Kubota's AD- and Lee's J-integrals. One reason for studying these integrals is that they may furnish interesting applications. For example, Sargent, Henstock and Lee ([8, Chapters 16 and 17]) established results in summability theory using one of these non­ absolute integrals. However, non-absolute integration is little used elsewhere. For simplicity of presentation we shall consider only real­ valued functions defined on the compact interval [a,b] . 1. Definitions of integration It is interesting to note that the Lebesgue integral does not include the ordinary integral in calculus. For example, consider the function (1) P(0) = 0 , F(x) = x 2sin 1/x2 , x £ 0 . Obviously, it is differentiable at every point on the real line. Hence in calculus we would say that the derivative F' as a function is integrable on the real line with F as its indefinite integral. However, F is not of bounded variation, for example, on [0,1] , and therefore is not absolutely continuous there. Thus its derivative F' is not Lebesgue integrable on [0,1] . The weakest known integral which includes both the Lebesgue integral and the integral in calculus is perhaps the Denjoy integral [17]. It was first defined by the Math. Chronicle 2(1973), 105-116. 105 French mathematician A. Denjoy in 1912. His original definition is lengthy and difficult. Here we give an equivalent definition due to Lusin (1912). A function F is said to be AC+ or absolutely continuous in the restricted sense on a setX if for every e > 0 there is 6 > 0 such that for every finite or infinite sequence °f non­ overlapping intervals with endpoints inX and J ml' <6 we have I u(F;.rn) < e , where ml denotes the length of I and w the oscillation of F n n over I , i.e . n w(F;-Tn ) = sup {|F (v) - F(u) | : u,v t J^} . A function F is said to be ACG* or generalized absolutely con­ tinuous in the restricted sense ona ,b[] if [a,&] is the union of a sequence of sets on each of which F is AC+ . For example, the function F in (1) is ACG* on [0,1] , though not AC+ there. Finally, a function is said to bespecial Denjoy integrable on [a,b] if there is a continuous function F which is ACG+ on [a ,b ] and whose derivative F '( x ) = f(x ) almost everywhere. Then the special Denjoy integral of f is given by fCC f = F(x) - F (a) , a 5 x < b . •'a The function F is uniquely determined except for an additive constant. The uniqueness of F is usually proved byveductio ad absurdum. A constructive proof following [13] is also possible. In 1957 Kurzweil [12] defined an integral of Riemann-type. A few years later Henstock gave independently the same integral, studied its properties, and showed that it is indeed equivalent to the special Denjoy integral. Following [9], we define Henstock's Riemann-complete integral as follows. Let 6(s) > 0 be a positive function on[a,b] . Then a division V , given by a = x < x < . < x = b with z . * 7 0 1 n J 106 being a point of [x. ,x .] for j = 1,2, n , is said to be J-l C compatible with 5(2) if for each j - 1>2, . n \X3 ' ^ < S(V ' i2i ' *.7-11 < St2J} ' The point z. is called the associated point of [x. ,x .] . Then a J v~ 3 function f is said to be Riemann-complete integvable on [a,b] with integral I if for every e > 0 there corresponds a function 6(s) > 0 on [a , b ] such that for all sums over divisions V of \a,b] compatible with 6(b) . The existence of divisions compatible with a given 6(s) > 0 is guaranteed by the Heine-Borel covering theorem. Henstock's definition differs from Riemann’s in that as 6(s) shrinks to 0 the intervals in the division do not necessarily shrink 'uniformly' . When 6(s) is a constant, we have the Riemann integral. There are other integrals which are equivalent to Denjoy's, for example, the Perron integral [17] and Henstock's variational integral [7]. But there are even more ways of defining the Lebesgue integral and, furthermore, some of them do not have analogues in the Denjoy case. One such example is a definition essentially due to F. Riesz [15]. That is, a fvnction f is Lebesgue integrable on [a,b] if and only if there exist real numbers , c2, ... and subintervals 1,1, ... of [a , b ] such that CO (2) I \ci \mli < - , i= 1 oo fix) = J e.ch(J.,x) almost everywhere, i= l 'l 'L where ml^ denotes again the length of _T. and ch(I^,x) the characteristic function of _Z\ . Then the Lebesgue integral of f 107 on [a>b~\ is given by rb (3) f = I o .ml. .L t ^ a PROBLEM 1. Giye a Riesz-type definition for the special Denjoy integral. In other words} relax the condition (2) so that we may still prove the uniqueness of the integral (3). In 1916 Khintchine and Denjoy [17] generalized independently the special Denjoy integral. The generalization is known as the general Denjoy or Denjoy-Khintchine integral. In 1931 J. C. Burkill {2] also gave a generalization of the special Denjoy integral, and called it the approximately continuous Perron integral or i4P-integral. Unfortu­ nately these two integrals do not include each other [19, p. 658]. A more general integral which includes both was given by Ridder [16, p. 148 Definition 7] and later independently by Kubota [10]. Kubota’s AD-integral is defined as follows. First, the approximate limit of a function F at a point denoted by (4) A = lim ap F O O is defined to mean: for every e > 0 the density of the set [x : A -e 5 F(x) 5 A + e) is unity at xQ . A function F is said to be approximately continu­ ous at xQ if (4) holds with A = F(xQ) . The approximate derivative of F at Xq , when it exists, is defined to be F(x) - F{x ) AD F (xn) = lim ap ---------- . 0 X ~ X0 Further, a function F is said to beAC or absolutely continuous in the wide sense on X if for every e > 0 there is6 > 0 such that for every finite or infinite sequence {[<2^,&^]} of non-overlapping intervals with endpoints a and b in J and ] \b -a I < 6 we r n n L 1 n n' 108 have I If(i„) - F(an)I < e . A function F is (ACG) on [a,2?] if [a,b] is the union of a sequence of closed sets on each of which F is AC . Note that every continuous function that is ACG* on [a,b] is also (.ACG) there. Then a function f is AD-integrable on [a,b] if there is an approximately continuous function F which is (ACG) on [a,b] and whose approximate derivative AD F(x) = f{x) almost everywhere. PROBLEM 2. Give a Riemann-type definition for Kubota's AD-integral. In other wordss show whether Henstock’s generalized Riemann integral [8] includes the AD-integral} and if nots define one that does. The so-called generalized Riemann integral defined by Henstock is an abstract form of the Riemann-complete integral. It includes many known integrals. I suspect that it also includes the 42?-integral. Naturally, another problem would be to define a Riesz-type AD-integral. 2. Convergence theorems The well-known dominated convergence theorem is in a sense the best possible result for the Lebesgue integral. It also holds for the special Denjoy [17] and ^-integrals [11]. THEOREM 1. Let f , n = 1, 2 , ..., g and h be special Denjoy or AD-integrable on [a,b] . If for n = 1, 2, ... f(x) = lim f (x) almost everywhere, n then f is special Denjoy or AD-integrable on [a,b] and rx a We should be able to improve the above theorem for Lee [14] proved 109 a convergence theorem which is not included in Theorem 1. We shall state the result as follows. A function f is J-integrable on [a,b] if there is a continuous function F whose derivative exists and is equal to f everywhere in [a,b] except perhaps at a countable number of points. Note that the function F is uniquely determined except for an additive constant. The J-integral is intermediate between the integral in calculus and Denjoy's. Hence every J-integra­ ble function is special Denjoy integrable. THEOREM 2. Let f j n = 1 j 2j ..., be J-integrable on [a,Z>] . Suppose that there -is a countable subset D of [asb ] such that the following conditions are satisfied: (i) for every x € ([a,&] - D) 3 there is a neighbourhood N{x) such that on N{x) - D the sequence ^fn ^ converges uniformly to a function f , (ii) for every x £ D there is a neighbourhood N(x) such that on N(x) the sequence {F' } of indefinite integrals of f^ converges uniformly} then f is J-integrable on [a3b] and we have (5). As in [14], consider the function r~n „ 0-n 2 , 0 < x < 2 0n _-n --n+i -2 , 2 < x < 2 Jf n O ) = 0 elsewhere The sequence {f^} is not dominated by any J- , special Denjoy or AD-integrable function on [0,1].