We consider real functions on the interval [0, 1]. Denote by Δ the set of derivatives; i.e., Δ = {ƒ:ƒ is a derivative}

= {ƒ : ∃F: [0,1]→ R {F is differentiable and ƒ = F')}.

If ƒ є Δ, any F with F' = ƒ  is a primitive of ƒ and is uniquely determined up to a constant. To normalize, we denote by F(x) = ʃ^x)0 ƒ the unique primitive of ƒ with F(0) = 0. This is the original Newtonian concept of integration as the inverse operation of differentiation, i.e., antidifferentiation

Kechris, Alexander S.

Caltech Authors - Main

Proceedings of the International Congress of Mathematicians 
Berkeley, California, USA, 1986 
The Complexity of Antidifferentiation, 
Denjoy Totalization, and Hyperarithmetic Reals 
ALEXANDER S. KECHRIS 
1. The problem of the primitive. We consider real functions on the in-
terval (0, 1]. Denote by tl. the set of derivatives; i.e., 
tl. = {! : f is a derivative} 
= {!: 3F: [0,1]--+ R {F is differentiable and f = F')}. 
If f E tl., any F with F' = f is a primitive of f 8Jld is uniquely determined up 
to a constant. To normalize, we denote by F(x) = J; f the unique primitive of 
f with F(O) = 0. This is the original Newtonian concept of integration as the 
inverse operation of differentiation, i.e., 8Jltidifferentiation. 
We will be concerned here with some definability aspects of the 
CLASSICAL PROBLEM OF THE PRIMITIVE. Reconstruct the primitive F of 
a given derivative f. 
This goes back to Newton 8Jld Leibniz 8Jld has been considered over the years 
in a different light as the concept of function has evolved (see [L2, P, S]). In 
"modern times" this problem was solved by Cauchy for continuous f (Cauchy 
definition of the integral) 8Jld Riemann for Riemann-integrable f (Riemann inte-
gration). Various generalizations were developed in the last half of the nineteenth 
century until Lebesgue introduced in his thesis {"Integrale, Longueur, Aire,, 
1902) his concept of integral. As explained in his book Lefions sur /'Integration 
et la Recherche des Fonctions Primitives [Ll], published in 1904, his primary 
motivation was the solution of the problem of the primitive in the general case 
of an arbitrary derivative. His Lebesgue integral made a major breakthrough 
here and totally resolved the problem in the case of a bounded derivative, more 
generally a Lebesgue integrable one. However, Lebesgue regretfully pointed out 
that he was unable to solve the problem completely for nonintegrable f. He 
considered this a major open problem that awaited solution. Such a solution 
was finally achieved in 1912 by A. Denjoy. The second edition of Lebesgue's 
aforementioned book [L2] published in 1928, contains an extensive treatment of 
Denjoy's work. 
Research partially supported by National Science Foundation Grant DMS-8416349. 
© 1987 International Congress of Mathematicians 1986 
307 
308 A. S. KECHRIS 
2. Denjoy totalization. In his solution to the problem of the primitive, 
Denjoy developed a concept of integration called Denjoy totalization. It con-
sists of a transfinite iteration of Lebesgue integrations, computations of limits 
of sequences, and summations of series. (It is worth noting that transfinite iter-
ation has also been used earlier in the context of Riemann as well as Lebesgue 
integration-even by Lebesgue himself.) We will briefly review now the concept 
of Denjoy totalization. 
We begin with the following result of Lebesgue. 
PROPOSITION 2.1. Let E ~ [a,b] be closed and {(ai,bi)} the intervals con-
tiguous to E in [a, b] (i.e., the components of its complement). Let F be differen-
tiable on [a, b] and assume F' is Lebesgue integrable onE and L IF(bi)- F(ai) I < 
oo. Then 
F(b)- F(a) = £ F'(x) dx + L[F(bi)- F(ai)]. 
This motivates the following definitions. 
DEFINITION 2. 2. Let E be a closed set and f a Lebesgue measurable func-
tion. We say that a point x E E is a point of nonsummability off on E if f is 
not Lebesgue integrable in every In E, I an open interval containing x. 
DEFINITION 2. 3. Let F be a continuous function on [a, b], let E ~ [a, b] be 
closed, and let {(ai, bi)} be the contiguous intervals of E in [a, b]. A point x E E 
-is called a point of divergence ofF onE if LI [F(bi)- F(ai)l = oo for all open 
intervals I containing x, where LI indicates that we include only the (ai, bi) 
contained in I. 
One has now the following fact. 
PROPOSITION 2. 4. IfF is a differentiable function on [a, b] and E C [a, b] is 
closed, then the points of nonsummability ofF' on E and the points of divergence 
ofF on E form a closed nowhere dense set in E. 
We can describe now the process of Denjoy totalization: 
Given f(= F') we reconstruct the primitive F-or actually the differences 
F(b) - F(a) for all a < b in [0, 1]-by transfinite induction as follows. We 
construct a decreasing transfinite sequence E1 2 E2 2 E3 2 · · · 2 Ea 2 · · · 
of closed subsets of [0, 1], each of which is nowhere dense in its predecessor (at 
successor stages), and simultaneously for each a the differences F(b)- F(a) for 
(a, b) disjoint from Ea. Since for some countable ao we must have Ea0 = 0, it 
follows that by stage a0 we have constructed F(b) - F(a) for all a < bin [0, 1]; 
i.e., we have reconstructed the primitive of f. 
To start with, we let 
E 1 = {x E [0, 1]: xis a point of nonsummability off on [0, 1]}. 
Let {(at,b})} be the contiguous intervals of E 1 in [0,1]. Then F(b)- F(a) is 
computed by Lebesgue integration for all a < b with [a, b] n E1 = 0, and thus 
by taking limits (since F is continuous) we can compute F(bJ) - F(aJ) as well. 
THE COMPLEXITY OF ANTIDIFFERENTIATION 309 
Now let 
E2 = {x E Et: xis a point of nonsummability of I on Et 
or x is a point of divergence of F on Et}. 
Note that this makes sense since we know F(bi) - F(a!). Then again, let 
{(a~, b~)} be the intervals contiguous to E2 in [0, 1]. Using Proposition 2.1 we 
can now compute F(b) -F(a) for all [a, b] disjoint from E 2 (letting E = Et n[a, b] 
there), by the formula 
F(b)- F(a) = 1 f(x) dx + L:[F(bD- F(aDJ. 
E [a,b] 
Then we compute F(bn - F(a~) by taking limits as before. 
We proceed this way by transfinite induction, taking intersections at limit 
ordinals. 
Of course the concept of arbitrary (countable) ordinals is essential in Denjoy 
totalization, and Denjoy as well as Lebesgue considered that a major application 
of the ideas of Cantor's transfinite set theory to analysis and therefore, in their 
view, an important justification of this theory. (Denjoy has, of course, repeatedly 
emphasized the relevance of transfinite set theory in analysis (see [Dl] and [D2]). 
Although Denjoy totalization employs only countably many ordinals for each 
given derivative I, Denjoy constructed examples of l's for which his process 
takes arbitrarily long countably many steps. In other words, the totality of all 
countable ordinals is necessary in his totalization. 
This problem was therefore raised: to what extent is the transfinite induction 
and the use of the totality of countable ordinals necessary in the problem of 
reconstructing the primitive (as opposed to its use in a particular process for 
doing that). See, for example, [P, pp. 170-171], which refers to Lusin's Thesis 
1915 as one of the first places where this question has been discussed. Other 
definitions of integrals (avoiding any use of ordinals) have been proposed, such as 
the Perron integral, the Kurzweil-Henstock integral, etc. (see [B]). These can be 
used to recover the primitive of any derivative, but they are hardly constructive 
in any sense. 
3. The complexity of antidifferentiation. The preceding discussion 
brings us to some recent results which address these problems from the point of 
view of logic and definability theory-more particularly, in this case, descriptive 
set theory. The idea is to classify the complexity of the operation of antidifferen-
tiation (*)I t-t J I. If one can show that this is sufficiently complex (not Borel as 
we explain in a moment), then this indicates that there is no simple constructive 
notion of integral (some kind of "super" Lebesgue integration) which is sufficient 
to invert any derivative. Thus, this shows the necessity of the use of transfinite 
induction over all the countable ordinals in any constructive such process. 
First we have to make precise in what sense we will express the complexity of 
(*),and this amounts to making clear in what sense a derivative f is considered 
310 A. S. KECHRIS 
as "given." Since every derivative is a Baire class 1 function, the most reasonable 
way to consider a derivative as "given" is via a "code" of it as a Baire class 1 
function, i.e., in terms of a sequence of continuous functions pointwise converging 
to it. (One could argue that giving such a sequence gives in some sense too much 
information about f. But since the point of the results below is that one cannot 
define simply the primitive of f, even with this extra information given, this is 
even better.) 
So let C[O, 1] denote the Polish space of continuous functions on [0, 1] with 
the uniform metric and let C[O, 1]N be the Polish space of infinite sequences of 
continuous functions with the product topology. Let 
CN = {7 E C[O, 1]N: Vx({fn(x)} converges)} 
be the class of pointwise converging sequences of continuous functions, and for 
7 E CN let f =lim 7 be defined by f(x) =lim fn(x) for all x E [0, 1]. It is not 
hard to verify that CN is a complete Tii subset of C[O, 1]N, so it is not Borel. 
Now let 
~ = {7 E C[O, 1]N: 7 E CN and lim7 E ~} 
be the set of codes of derivatives. The first result here was proven by M. Ajtai 
several years ago and provides an upper bound for the complexity of ~. 
THEOREM 3.1 (AJTAI, UNPUBLISHED). The set~ is IJi. 
The following lower bound completes the classification. 
THEOREM 3.2 [KJ. The set~ is not EL i.e., there is no Ei setS~ CN 
such that for 7 E CN, 7 E S <=? 7 E ~. The same holds even for b1~ = 
{7 E C N: lim 7 is a derivative of absolute value :$ 1}. 
This computes the complexity of the domain of the operation of antidifferen-
tiation. For the operation itself one first has the following upper bound. 
THEOREM 3.3 (AJTAI, UNPUBLISHED). The operat~on of antidifferentia-
tion is ~i. More precisely there are Ei, Tii relations S, P ~ C[O, 1JN x R 2 
such that for 7 E CN, with f = lim7 E ~ and all x E [0, 1], y E R, we have 
y <fox f <=? S(f, x, y) <=? P(f, x, y). 
Notice that by standard effective descriptive set theory Theorem 3.3=>Theo-
rem 3.1. Ajtai's proof of Theorem 3.1 and 3.3 used nonstandard models and 
Denjoy totalization (oral communication). An alternative way to prove Theo-
rem 3.3 is to show that the transfinite process in the Denjoy totalization for f 
terminates at some ordinal recursive in 7 (where lim7 = !). This approach uses 
also a crucial boundedness argument due to H. Woodin. 
On the other hand, it was recently established that antidifferentiation is not 
Borel. 
THE COMPLEXITY OF ANTIDIFFERENTIATION 311 
THEOREM 3. 4 (DOUGHERTY- KECHRIS [DK]). The operation of antidif-
ferentiatz'on is not Borel. More precisely, there is no Borel set B ~ C[O, 1]N such 
that for 1 E CN, with f =lim 1 E .6., 1 E B <=> I~ f > 0. 
This shows that the totality of countable ordinals is necessary in any construc-
tive process for recovering the primitive, since no "simple analytical" operation 
suffices for this purpose. We take here as a necessary characteristic of such an 
operation that it is Borel (in the codes). This is clearly the case for Riemann or 
Lebesgue integration, taking of limits of sequences, summation of series, etc. So 
one can argue that arbitrary countable ordinals are intrinsically connected with 
the operation of recovering the primitive itself, and not only with a particular 
process of reconstruction, like Denjoy's. 
4. New characterizations of the hyperarithmetic reals. What is behind 
the preceding results is actually a new characterization of the hyperarithmetic 
reals. A sequence 1 E C[O, 1]N is called recursive if it is a recursive sequence of 
recursive functions. Then Theorem 3.3 can be rephrased as follows. 
THEOREM 4.1 (AJTAI, UNPUBLISHED). Let 1 E C[O, 1jN be recursive, 1 E 
CN, f =lim 1 E .6.. Then the primitive If is D.i. (Similarly relativized to any 
given real.) 
Note that F = If is by its definition a TI~ singleton (iff= lim 1, 1 recursive). 
That it is actually D.i is based ultimately on the Denjoy totalization, so this gives 
a definability consequence of this process, which clearly quantifies its constructive 
aspects. 
Recall that a real x E R is hyperarithmetic (or D.D if { r E Q: r < x} is 
hyperarithmetic. Call also a derivative f E .6. recursive if there is recursive 
1 E CN, with f =lim f. Then one has immediately from Theorem 4.1 that for 
a recursive derivative j, Ig f is a hyperarithmetic real (and similarly relativized 
to any given real). The main result of [DK] is the converse to this. 
THEOREM 4.2 (DOUGHERTY-KECHRIS [DK]). Let x E R. Then the fol-
lowing are equivalent: 
(i) x is hyperarithmetic, 
(ii) x =I~ f, for some recursive derivative f. 
(Similarly relatz'vised to any given real.) 
The proof of this theorem involves effective transfinite induction based on the 
Recursion Theorem. 
One can further combine Theorem 4.2 with Matyasevich's Theorem to ob-
tain formulas and an equivalent characterization of the hyperarithmetic reals 
involving only classical notions of analysis. 
312 A. S. KECHRIS 
THEOREM 4.3 (DOUGHERTY-KECHRIS [DK]). The hyperarithmetic reals 
are exactly those of the form J~ J, where f(x) is a derivative given by 
00 max (0, 1-1 (n- [y'n)2 + 1) X- (rv'nJ- ([V'nJ2 ) )I) 
f(x) = L an 2 , 
n=O n- [y'n] + 1 
with 
max(O, 1- Q(n, m1 · · · mN )2), 
o:::;ml, ... ,mN:::;222 N 
and Q an exponential polynomial with coefficients in Z. 
Another version of this kind of result can be stated as follows: call a function 
analytically expressible if it can be expressed by an explicit formula involving 
elementary functions and I:~=o· Then one can show that the hyperarithmetic 
reals are exactly the reals of the form J01 cp, where cp is an analytically expressible 
derivative. This demonstrates clearly the "definability gap" between a function 
and its derivative. One can have a derivative given by a simple analytical formula, 
while its primitive is immensely complex. 
5. Some open problems. (A) The first problem is related to definability 
aspects of so-called "descriptive definitions of integrals" (see [S, Chapters VII, 
VIII]). These are essentially implicit definitions like the original one of the prim-
itive. For example, the Lebesgue integral F of an integrable function f can be 
defined as the unique (up to a constant) F which is such that (i) F is absolutely 
continuous, and (ii) F'(x) = f(x) for almost all x. By replacing, in (i), absolute 
continuity by more general conditions, one can obtain descriptive definitions of 
integrals inverting any derivative. The question is whether these conditions can 
possibly be Borel. (Note that (ii) is clearly Borel and so is the condition of 
absolute continuity.) This leads to the following 
PROBLEM. Is there a Borel relation B ~ C[O, 1]N x C[O, 1] such that if 
f E CN, f =lim/ E ~.and FE 0[0,1], then F = f f #(/,F) E B. We 
conjecture that the answer is no. (This would be stronger than Theorem 3.4.) 
(B) An interesting but a bit vague problem is to come up with simpler formulas 
in Theorem 4.3. Another more precise question is the following. If in Theorem 
4.3 we expand each summand off in a Fourier series, then we obtain a formula 
for the hyperarithmetic reals of the form J~ E~=O E:=o anm cos m(x-bn), with 
explicitly given anm, bn. Can we express them simply as 
{1 00 
Jo ~an cos n(x- bn), 
with recursive {an}, {bn}? 
(C) The final problem is related to another important result of Denjoy. If a 
27r-periodic function f(x) has a trigonometric expansion 
00 
f ( x) = L (an cos nx + bn sin nx), 
n=O 
THE COMPLEXITY OF ANTIDIFFERENTIATION 313 
then the coefficients {an}, {bn} are uniquely determined by Cantor's Uniqueness 
Theorem. Iff is Lebesgue integrable, the an, bn can be computed by Fourier's 
formula. But how does one find an, bn from fin the general case? This problem 
was solved by Denjoy in 1921 (see [Dl]) by an extremely complicated procedure, 
again involving transfinite induction. From the definability point of view this 
leads to the following two questions: 
PROBLEM. Classify the complexity ofT = {] E C N: lim 7 admits a trigono-
metric expansion}. 
PROBLEM. Classify the complexity of the operation 7 1-t {an}, {bn}, where 
7 E T, lim]= f = L:(ancosnx + bn sinnx). 
From its definition T is E~ and this operation is 6.~ on T. However, it 
appears that Denjoy's constructive process for recovering an, bn ought to lead 
to a substantial lowering of this complexity, perhaps at the level of Ei or Tii 
nonmonotone inductive definitions. If this is correct and if one can show corre-
sponding lower bounds, this would lead to a classification of a natural and basic 
concept of analysis which falls between two levels of the projective hierarchy. 
This certainly would be a very interesting phenomenon. Also, if it is indeed 
true that the complexity of computing the trigonometric expansion is quite high 
above Borel, this would give a nice definability "explanation" of the considerable 
difficulty of Denjoy's procedure. At this point, however, this is only speculation, 
since it is not even known whether 7 1-t {an}, {bn} is Borel or not. 
REFERENCES 
[B] P. Bullen, Non-absolute integrals: A survey, Real Anal. Exchange, 5 (1979-80), 195-
259. 
[Dl] A. Denjoy, Le~ons sur le calcul des coefficients d'une aerie trigonomt!trique, I-IV, 
Gauthier-Villars, Paris, 1941-49. 
[D2] __ , L't!numt!ration transfinie, Gauthier-Villars, Paris, 1946. 
[DK] R. Dougherty and A. Kechris, The complexity of antidif!erentiation, Nov. 1985; 
Addendum, Analytical expressions for the hyperarithmetic reals, Feb. 1986; mimeographed 
circulated notes. 
[K] A. Kechris, Classifying the complexity of the set of derivatives, July 1985, mimeo-
graphed circulated notes. 
[Ll] H. Lebesque, Le~ons sur l'intt!gration et Ia recherche des fonctions primitives, 
Gauthier-Villars, Paris, 1904. 
[L2] __ ,ibid., 2eme ~d., Paris, Gauthier-Villars, 1928. 
[P] I. Pesin, Classical and modern integration theories, Academic Press, New York, 1970. 
[S] S. Saks, Theory of the integral, 2nd rev. ed., Dover, 1964. 
CALIFORNIA INSTITUTE OF TECHNOLOGY, PASADENA, CALIFORNIA 91125, USA 


The Complexity of Antidifferentiation, Denjoy Totalization, and Hyperarithmetic Reals

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