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 applica­tions. 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 non­absolute integrals. However, non-absolute integration is little used elsewhere. For simplicity of presentation we shall consider only real­valued 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) = x2sin 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 d ifficult . 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 set X if for every e > 0 there is

6 > 0 such that for every finite or infinite sequence ° f non­

overlapping intervals with endpoints in X and J ml' < 6 we have

I u(F;.rn) < e ,

where ml denotes the length of I and w the oscillation of Fn 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 on [a ,b ] i f [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 be special 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 by veductio 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

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being a point of [x. ,x .] for j = 1,2, n , is said to beJ-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 aJ v~ 3

function f is said to be Riemann-complete integvable on [a,b] withintegral I if for every e > 0 there corresponds a function6(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 subintervals1 , 1 , ... of [a,b] such that

CO(2) I \ci \mli < - ,

i= 1oo

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

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rb

on [a>b~\ is given by

(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 FO 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 x Q . A function F is said to be approximately continu­

ous at x Q if (4) holds with A = F (x Q) . 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 be AC or absolutely continuous in

the wide sense on X if for every e > 0 there is 6 > 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 wer n n L 1 n n'

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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, ...

We should be able to improve the above theorem for Lee [14] proved

f(x) = lim f (x) almost everywhere, n

then f is special Denjoy or AD-integrable on [a,b] and

rx

a

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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-integra­ble 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

f O) =J n

r~n „ 0-n2 , 0 < x < 20n _-n --n+i-2 , 2 < x < 2

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.

Dieudonnd [5] defines a Kothe space as follows. Let L be the space of all Lebesgue integrable functions on [a,&] , and M a sub­set 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 varia­tion 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 fromD to L . For 0 < a < 1 , a function F is said to be in W ifa

3. Kothe dual spaces

(6) sup j \FixJ - Fix. )|1/a =̂l 1

where the supremum is taken over all divisions a < x ^ < x ^ < ...< x = b . For convenience, denote by W the class of all boundedn 0measurable functions on [a,£>] . Further, a special Denjoy integrable function / is said to be in D if its indefinite special Denjoy integral F is in . Note that W is simply the space BV of all functions of bounded variation, D the space L of all Lebesgue 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

1/a11/IL = supa

an

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

rbL(f) = fg •

Ja

When a = 1 , the result is well known. When a = 0 , it is due

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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 follow­ing 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. Representa­tion 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) =

where

K(f(x) ,x)dxa

(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), 2 1 0 - 2 2 0 .

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 L s p a c e s 3 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.

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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].

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