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]. Yet, applying Theorem 2 we have (5) with /(x) being zero.
PROBLEM 3. State and prove a convergence theorem for the special Denjoy or AD-integrals such that it includes both Theorems 1 and 2.
Glancing through certain proofs involving non-absolute integrals, for example [18], one feels that the proof may be made simple through a suitable convergence theorem. 3. Kothe dual spaces
Dieudonnd [5] defines a Kothe space as follows. Let L be the space of all Lebesgue integrable functions on [a,&] , and M a sub set of L . A Kothe space N with defining set M is the space of all measurable functions f such that the product fg is Lebesgue integrable on \a,b\ for all g £ M . The Kothe dual space of N is the Kothe space with defining set N . For example 3 1 5 p < 00 is a Kothe space and its dual is the space L^ where 1/p + 1/q = 1 . Similarly, we may define Kothe spaces with the Lebesgue integral replaced by the special Denjoy. Now let D be the space of all special Denjoy integrable functions on [a,2?] , and EBV the space of all functions each of which is equal to a function of bounded varia tion almost everywhere on [a,b] . Sargent [18] showed that the Kothe dual of D is EBV .
In general, let us first define a scale of function spaces from D to L . For 0 < a < 1 , a function F is said to be in W if a
(6) sup j \FixJ - Fix . )|1/a ^=l 1 where the supremum is taken over all divisions a
integrable functions, and D q the space D of all special Denjoy integrable functions. The following generalization is due to Burkill and Gehring [3].
THEOREM 3. For 0 5 a 5 1 , the product fg is special Den joy integrable on [a,b] for every f € D if and only if g is almost everywhere equal to a function in W
The following question remains open.
Ill PROBLEM 4. Let AD be the space of alt AD-integrable functions on [a,b] and J the space of all J-integrable functions on [a,b] .
Find the Kb the dual spaces of AD and J .
The next problem does not involve non-absolute integration. I include it here for its own interest.
PROBLEM 5. Let L , 0 < p < 1 , be the space of all measurable functions such that ^\f\^ is Lebesgue integrable on [a,&] . Find the Kothe dual space with defining set L^ , 0 < p < 1 .
Note that L^ , 0 < p < 1 , is a Frechet but not a Banach space. Day [4] showed that there exists no non-trivial continuous linear functional on L . However, we may consider 'additive functionals’ on the space L. We shall return to this in the next section.
4. Representation theorems
Let , 0 5 a 5 1 , be defined as in section 3. Following (6), we can introduce a norm in ^ , 0 < a < 1 , to be a n 1/a 11/ILa = sup t=l where the supremum is taken over all divisions a < x Q < < ... < x = b . When a = 0 , the norm jlfll is defined to be n 0 w(F, [o.,b]) , the oscillation of F over [a,b] . Hence we can define continuous linear functionals on D. Burkill and Gehring [3] proved the following representation theorem.
THEOREM 4. Suppose that 0 5 a 5 1 and that L(f) is a continuous linear functional defined on D . Then there exists a function g in W ^ such that for each f € D^ the product fg is special Denjoy integrable on [a,b\ and rb L(f) = fg • Ja When a = 1 , the result is well known. When a = 0 , it is due
112 to Alexiewicz [1]. Let L , D and AD be defined as in section 3. A natural question will be to ask if we can define a scale of function spaces from L or D to AD and prove a representation theorem for continuous linear functionals on these spaces. Evidently, it will depend on the solution of Problem 4 above.
An operator T defined on a function space M into another function space N is said to be additive if it satisfies the follow ing conditions:
(7) For every e > 0 and b > 0 there exists 6 = 5(e,£>) > 0 such that for f , g € M with ||/|| 5 b , ||^|| 5 b and ||/-^|| 5 6 we have ||2*/ - Tg\\ 5 e ;
(8) for every b > 0 , there exists B = B[b) > 0 such that for f € M with ll/H < b we have llT/ll 5 B , i.e.
sup {||T/|| : ll/H < M < ® ;
(9) if f(x)g(x) = 0 for almost all x in [a,b] , then
T(f + g) = Tf + Tg .
When N is the set of real numbers, T is called an additive functional. Note that an additive functional need not be linear. Therefore conditions (7) and (8) above are not equivalent. Representa tion theorems for additive functionals on the spaces L^ , 0 < p < °° , have been studied by Friedman and Katz [6]. Further, Woyczynski [20] extended the results of Friedman and Katz to Orlicz spaces. We shall state the theorem of Woyczynski in special cases. First, a function K(y,x) defined for -°° < y < °° and a 5 x 5 b is said to satisfy the Caratheodory conditions if it is continuous in y for almost all x in [a,b] and measurable in x for each fixed y . For a given K(y,x) , let K denote the operator defined by Then we have
THEOREM 5. L(f) is an additive functional defined on L ,
0 < p < °° , if and only if
L(f) = K(f(x) ,x)dx a where (i) • K(0,x) = 0 ; (ii) K(y,x) satisfies the Caratheodory conditions;
(iii) for -°° < y < °° and a 5 x 5 b the inequality
\K{y,x)\ 5 ayP + g(x) holds with some non-negative constant a and some Lebesgue integrable function g ; (iv) K is an additive operator on L .
We remark that when p > 1 , Theorem 5 reduces to the standard representation theorem for continuous linear functionals with K(f[x)jX) = f(x)h(x) for some h € , 1/p + 1/q = 1 . The following problem, as far as I know, has never been studied.
PROBLEM 6. State and prove representation theorems for additive functionals on the space D of special Denjoy integrable functions and, in general} the spaces D^ 0 < a 5 1 .
Furthermore, representation theorems for linear and nonlinear operators on D have not been fully investigated.
REFERENCES
1. A. Alexiewicz, Linear functionals on Denjoy integrable functions> Colloq. Math. 1(1948), 289-293.
2. J. C. Burkill, The approximately continuous Perron integrals Math. Z. 34(1931), 270-278.
3. J. C. Burkill and F. W. Gehring, A scale of integrals from Lebesgue’s to Denjoy's3 Quart. J. Math. Oxford (2) 4(1953),
210- 220 . 4. M. M. Day, The spaces Lp with 0 < p < 1 , Bull. Amer. Math. Soc. 46(1940), 816-823.
5. J. Dieudonne, Sur les espaces de Kothes J. Analyse Math. 1(1951), 81-115.
6. N. Friedman and M. Katz, Additive functionals on Lspaces3
Canad. J. Math. 18(1966), 1264-1271.
7. R. Henstock, Theory of Integration3 Butterworths, London, 1963.
8. R. Henstock, Linear Analysis3 Butterworths, London, 1967.
9. R. Henstock, A Riemann-type integral of Lebesgue power3 Canad. J. Math. 20(1968), 79-87.
10. Y. Kubota, An integral of the Denjoy type I3 II3 III3 Proc. Japan Acad. 40(1964), 713-717; 42(1966), 737-742; 43(1967), 441-444.
11. Y. Kubota, An abstract integral3 Proc. Japan. Acad. 43(1967), 949-952.
12. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter3 Czechoslovak Math. J. 7(82) (1957), 418-446.
13. P. Y. Lee, A nonstandard example of a distribution3 Amer. Math. Monthly 77(1970), 984-987.
14. P. Y. Lee, A nonstandard convergence theorem for distributions3 Math. Chronicle 1(1970), 81-84.
15. P. Y. Lee, The uniqueness of a Riesz-type definition of the Lebesgue integral3 Math. Chronicle 1(1970), 85-87.
16. J. Ridder, Uber die gegenseitigen Beziehungen verschiedener approximativ stetiger Denjoy-Perron Integrale3 Fund. Math. 22(1934), 136-162.
17. S. Saks, Theory of the Integral3 Second revised ed., English translation. Monografie Matematyczne VII, Warsaw, 1937.
115 18. W. L. C. Sargent, On the integrability of a product, J. London Math. Soc. 23(1948), 28-34.
19. G. Tolstoff, Sur I’integrale de Perron, Mat. Sb. 5(1939), 647-659.
20. W. A. Woyczynski, Additive functionals on Orlicz spaces} Colloq. Math. 19(1968), 319-326.
Nanyang University Singapore
Note added in proof 29 January, 1973 Problem 5 has been solved by Ng Peng Nung of Nanyang University. The Kothe dual space with defining set L > 0 < p < 1 , is trivial, i.e. y it consists of only the null function. The result is in agreement with that of Day [4], though its proof is independent of [4].
116