Part VI Fundamental theorems of integral on R Introduction

The Riemann integral can be considered an evolution of Cauchy’s integral, in that certain functions that were not integrable according to Cauchy became integrable in Riemann’s theory. At the same time, alas, in the new framework integration is no longer the inverse operation to differentiation. As a matter of fact, there exist Riemann ( ) = x ( ) integrable functions f whose integral map F x a f t dt is not differentiable (not even on one side only) at certain points. Thus the fundamental theorem of calculus, in the version for continuous maps proved by Cauchy, loses its status of calculus’ highest pinnacle and becomes a mere special case of a much bigger picture. Realising this fact led mathematicians to view differentiation and integration with a more critical eye, and prompted them to explore the depths of the relationship between continuity, differentiability and integrability. was one among many academics trying to unravel the connection differentiability–continuity. In 1871 Hankel used Riemann’s integral function1 R(t) to define the map  x G(x) = k + R(t)dt, a which is continuous everywhere, but not differentiable at infinitely many points. The example raised much incredulity among the scientific community, for it went against the conventional wisdom that continuity implied differentiability but at a finite number of points. Many textbooks of the time, to prove that a continuous map f (x) on some interval had to be differentiable, used geometrically-flavoured arguments involving tangents to the curve representing f (x), without fathoming that there do exist continuous maps not allowing for a pictorial description of that kind. It was Karl Weierstraß who cleared away all doubts when he constructed a function of one real variable that is everywhere continuous but never differentiable.

1Here R(t) will denote the function Riemann employed to show his notion of integral was sophis- ticated enough to allow the integration of maps with infinitely many discontinuities.

348 Fundamental theorems of integral calculus on R 349

Liberating the concept of derivative from continuity allowed to study differenti- ation independently, and thus extend its scope to discontinuous functions as well. For this reason Ulisse Dini, to make derivatives and integrals both rigorous and more general, and encompass maps with dense discontinuity sets (which he called “discon- tinuous dot-functions”2) introduced in the early 1880s the so-called Dini derivatives. These numbers were created to extend the notion of derivative to pathological functions. Explicitly, the (upper and lower) right Dini derivatives of F at a point x ∈[a, b) are

+ F(x + h) − F(x) F(x + h) − F(x) D F(x) = lim sup and D+ F(x) = lim inf ; → + h→0+ h h 0 h the (upper and lower) left Dini derivatives at x ∈ (a, b] are

− F(x) − F(x − h) F(x) − F(x − h) D F(x) = lim sup and D− F(x) = lim inf . → + h→0+ h h 0 h

+ − + Note that D F(x) ≥ D+ F(x) and D F(x) ≥ D− F(x). When D F(x) = − D+ F(x) = D F(x) = D− F(x) ∈ R the map F is differentiable at x and its derivative is f (x), precisely the common value of the four Dini derivatives. Among other things, these new concepts allowed Dini to state the fundamental theorem of calculus for the Riemann integral in the following form: if F is continuous and one of Dini’s derivatives DFis Riemann integrable, then the integral map F(x) = x DF(t)d(t) has all four Dini derivatives bounded and integrable, and moreover a b ( ) = ( ) − ( ) a DF x dx F b F a . Dini, moreover, explicitly emphasised the necessity of assuming the integrability of DF, and conjectured the existence non-constant, continuous primitives F that become constant over a suitable subinterval of any given interval. This would imply DF is not integrable, because we would not be able to enclose its discontinuities using a finite number of arbitrarily small intervals. By the way, the surmise was proved in 1881 by , a student of Dini at Scuola Normale Superiore in , who gave an explicit example (see p.xxx ff.). The existence of continuous maps with no derivative, and the ensuing demise of the fundamental theorem of Cauchy’s integral calculus, persuaded many mathemati- cians, most notably Lebesgue, to investigate the relationship between integrals and primitives. The first result of Lebesgue’s theory is that

“If f ∈ L1[a, b] the integral map  F(x) = f (t)(dt) [a,x] is continuous and has derivative F (x) = f (x) a.e. on [a, b].”

2‘funzioni punteggiate discontinue’ in Italian. 350 Fundamental theorems of integral calculus on R

Lebesgue also proved, conversely,

“If F :[a, b] → R has bounded derivative f , then f is Lebesgue integrable and the equality  F(x) − F(a) = f (t)(dt) [a,x] holds everywhere on [a, b].”

These results entitled him to declare in 1902 that the problem of integral calcu- lus, conjectured by Dini and proved by Volterra within Riemann’s set-up, had been completely solved in the case of bounded derivatives. However, things get more complicated if the derivative f is not bounded, for then f might not be integrable. Lebesgue tried to pin down the features a primitive F should have in order for f to be well defined almost everywhere on [a, b] and integrable on the entire interval. Restricting to the case where one Dini derivative of F is almost everywhere finite, Lebesgue showed that F must have bounded variation. It is at this point that the notion of bounded variation, for which we will provide a brief account, entered the stage. The idea was elaborated by Jordan for other reasons, namely in relationship to conditions granting that a function F(x) has a Fourier converging to F(x). Jordan proved a sufficient condition for convergence in terms of a function with bounded variation. To define bounded variation he took a map F :[a, b] → R and fixed a finite partition P ={a = x0, x1,...,xn−1, xn = b} of [a, b]. He then called variation of F(x) over [a, b] with respect to P ∈ P the quantity

n b( , ) = | ( ) − ( )|. Va P F F xi F xi−1 i=1

| ( ) − ( )| = ,..., The difference F xi F xi−1 , i 1 n, represents the largest variation of [ , ] n | ( ) − ( )|= { } F over xi−1 xi .If i=1 F xi F xi−1 k, the set of values k P∈P can, as P varies, have either finite or infinite least upper bound. Let us call total variation of F on [a, b] the supremum. When that is finite we say F has bounded variation on [a, b]:   b( ) = b( , ) : [ , ] ≤∞. Va F sup Va P F P partition of a b P∈P

A function with bounded variation is bounded, whereas there exist bounded maps whose variation is not bounded. To see this just take the Dirichlet function f : [0, 1] → R: albeit bounded, since f (x) ∈{0, 1}, it has oscillations of width 1 for every x ∈ Q ∩[0, 1]. Therefore functions with bounded variation form a proper Fundamental theorems of integral calculus on R 351 subfamily of bounded maps. Moreover, there exist discontinuous maps with bounded variation, such as discontinuous monotone functions, and there also are continuous maps with unbounded variation. But going back to the issue, Jordan proved that a function with bounded variation on [a, b] can be written as difference of two increasing functions (Cours d’analyse de l’École Polytechnique, first edition in three volumes published in 1882-1887). On the matter Lebesgue observed that if f ∈ L1 is positive, the integral map  F(x) = f (t)(dt), x ∈[a, b] [a,x] is increasing. Moreover, since a summable function f is the difference of positive summable functions f +, f −, any integral map is the difference of increasing func- tions

F(x) = F +(x) − F −(x).

Based on this, in 1904 Lebesgue established the following:

if F :[a, b] → R has bounded variation, then it is differentiable a.e. on [a, b], and f is summable on [a, b]. He stopped short of saying  |F(x) − F(a)|= | f (t)|(dt) [a,x] for every x ∈[a, b], because in that case the difference of the two sides would be a monotone map G with bounded variation and zero derivative -almost everywhere on [a, x].  ( ) =  ≥ The point is that Lebesgue viewed F x [a,x] fd (clearly for f 0) not as afunctionofx, but rather as being defined on an interval, or more generally as a set function μF : S(X) →[0, ∞], where 

μF (E) = fd, E ∈ S(X) E

designates a measure on the measurable space (X, S(X)). In practice μF is a weighted measure (weighted by the non-negative measurable map f ) determined by the for- mula dμF = fd.

Now the question is: is there a way to characterise, among all set functions μF ([a, b]), indefinite integrals that satisfy 352 Fundamental theorems of integral calculus on R

 F(x) − F(a) = f (t)(dt) [a,x] for every x ∈[a, b] ? The answer is yes, there is indeed, and it relies on two measure- theoretical notions that are central to the whole of : complete additivity and absolute continuity with respect to a given measure. The punchline is this: among functions with bounded variation, the only ones satisfying  F(x) − F(a) = f (t)(dt) [a,x] must be both • completely additive – in the words of De la Vallée Poussin:

“Given E ⊆ X measurable, a set function μF (E) is completely additive if, irre- spective of how one partitions E in a finite/countable number of measurable subsets Ei ,  ∞ ∞  μF (E) = μF Ei = μF (Ei ) ; i=1 i=1

• absolutely continuous, as defined by the mathematician Giuseppe Vitali thus:

“Given E ⊆ X measurable, a set function μF (E) is absolutely continuous with respect to measure  if μF (E) tends to 0 as (E) tends to 0.” Lebesgue went on to prove that

“If a set function with bounded variation μF ([a, b]) is absolutely continuous with respect  and completely additive, then it is differentiable a.e. on [a, b] (F  = f a.e. on [a, b]), its derivative f is integrable on [a, b], and for every x ∈[a, b]  F(x) − F(a) = f (t)(dt). [a,x]

Hence, F is the indefinite integral of f , and, in turn, f is the derivative of F a.e. on [a, b]. Where the derivative f is not defined it can take any value, since the set on which f does not exist has zero measure. The main tool in the proof of this important result is Vitali’s covering theorem. Once we have found a set function with bounded variation μF (E) satisfying the above conditions, we would like to know how to identify the f with indefinite integral μF (E). The procedure for finding f starting from μF (E) generalises the method used to differentiate. Suppose we wish to find f at a point x0 ∈ E. We compute μF on a ball B(x0, r) centred at x0, the measure (B(x0, r)) and then the ratio   μ (B(x , r)) ( , ) fd F 0 = B x0 r . (B(x0, r)) (B(x0, r)) Fundamental theorems of integral calculus on R 353

From the complete additivity and absolute continuity of μF with respect to , by letting (B(x0, r)) → 0 the ratio converges to the value of the non-negative, measurable map f at x0, where f is precisely the derivative at x0 of μF (E): μ (B(x , r)) dμ f (x ) = lim F 0 = F (x ). 0 → 0 r 0 (B(x0, r)) d

Clearly, in case the limit does not exist one has to resort to the Dini derivatives, which generalise the derivative. This is not the right place to discuss the existence of the derivative f of μF , which requires tools – such as the Radon-Nikodym theorem – that will be dealt with appropriately in Volume Two. Despite the enormous advancements made in the pursuit of primitives, it must be said that the Lebesgue integral does not settle the matter once and for all. There are situations in which f is neither Riemann integrable nor Lebesgue integrable, and yet μF (E) exists and has f as derivative -almost everywhere on E. To find these primitives we need a novel idea, and a broader concept of integral. The new integral was fleshed out by Denjoy and Perron, who independently developed the same theory, starting from separate premises and different viewpoints. The technique devised by Denjoy, called complete totalisation, is constructive: it starts from the Lebesgue integral, and defines a hierarchy (finite or transfinite) of operations meant to determine explicitly the primitive of any derivative. Complete totalisation is the inverse process to differentiation in the aforementioned generalised sense. Denjoy’s approach combines Lebesgue’s analysis with the transfinite methods used by Baire, and as it happens, it is completely analogous to the method of Baire’s 1899 PhD thesis.