22. Integration in locally compact spaces.

The development of the modern notion of integral is tightly linked to the evolution of the idea of function, and to a deep study of the numerical functions of real variables, which has been taking place since the beginning of the XIXth century: It is known that Euler already conceived of the notion of function in a fairly general way, since for him a given "arbitrary" curve intersected in a single point by every parallel to the axis $Oy$ defined a function $y = f(x)$ (cf. p. 196); but, like the majority of his contemporaries, he refused to allow that such functions could be expressed "analytically". This point of view was to change hardly at all until the work of Fourier; but the discovery, by this latter, of the possibility of representing discontinuous functions as sums of trigonometric series,$^1$ was to exercise a decisive influence on subsequent generations. Truth to say, the proofs of Fourier totally lacked in rigour, and their domain of validity did not appear clearly; however the integral formulae

giving the coefficients in the expansion of $\phi$ as a Fourier series, had an intuitively obvious meaning as soon as $\phi$ is assumed to be continuous and piecewise monotonic.$^2$ Also it is at first to these functions that Dirichlet limits himself, in the famous memoir ([92], v. I, pp. 117-132) where he established the convergence of Fourier series; but already, at the end of his work, he is concerned with the extension of his results to wider classes of functions. It is

$^1$ It is not a case besides of "discovery" except in a totally relative sense: Euler knew already about expansions in trigonometric series of non periodic functions such as $x$ or $x^2$, and the formulae (22.1) are to be found in a work of Clairvaux already in 1754 and with Euler in a memoir of 1777. But where the XVIIIth century, due to the lack of a clear conception of what is meant by an expansion in series, neglected such results and kept intact the belief in the impossibility of obtaining such expansions for "discontinuous" functions, Fourier, on the contrary, proclaims that these expansions are convergent whatever the given curve may be which corresponds to $\phi(x)$, whether it can be assigned an analytic equation, or whether it does not depend on any regular law ([112], v.I, p. 210).

$^2$ For Fourier, the integral is still defined by appealing to the notion of area; the analytic definition of the integral does not appear, let us remember, until Cauchy (cf. p. 197).

known that it is on this occasion that Dirichlet, making the ideas of Fourier precise, defines the general notion of function such as we understand it today; the first point to clarify was naturally to know in which cases it was still possible to attach a meaning to the formulae (22.1). "When the solutions for the continuity [of $\phi$] are infinite in number ..." says Dirichlet (loc. cit., pp. 131-132), "it is necessary that then the function $\phi(x)$ should be such that, if one denotes by $a$ and $b$ two arbitrary quantities between $-\pi$ and $+\pi$, it is always possible to put between $a$ and $b$ other quantities $r$ and $s$ close enough so that the function remains continuous in the interval $r$ to $s$. The necessity for this restriction will be felt by considering that the different terms of the series [of Fourier] are definite integrals and in going back to the fundamental notion of integrals. It will be seen then that the integral of a function does not mean anything except in as much as the function satisfies the condition previously stated."

In modern terms, Dirichlet seems to believe that integrability is equivalent to the fact that the points of discontinuity make up a sparse set; he points out besides, a few lines later, the famous example of the function equal to $c$ for $x$ rational, and to a different value $d$ for $x$ irrational, and affirms that this function "could not be substituted" in the integral. He announced besides subsequent work on this subject, but this work was never published,$^3$ and for 25 years, no one seems to have sought to advance along this path, perhaps because the consideration of such "pathological" functions seemed at the time completely lacking in interest; in any case, when Riemann, in 1854 ([259 a], pp. 227-264), takes up the question (always in connection with trigonometric series$^4$) he experiences the need to justify his work: "Whatever our ignorance concerning the way in which the forces and the states of matter vary with time and place in the infinitely small, we can nonetheless take it as certain that the functions to which the researches of Dirichlet do not apply, do not occur in natural phenomena. However", he continues, "it appears that these cases not treated by Dirichlet deserve attention for two reasons. Firstly, as Dirichlet himself remarks at the end of his work, this subject is in a very close relation to the principles of the infinitesimal calculus, and can be of use to bring more clarity and certainty to these principles. From this point of view, his study has an immediate interest. In the second place, the application of Fourier series is not limited to research in physics; they are at present applied also with success in a domain of pure mathematics, number theory, and there it seems that they are precisely the functions whose expansion in trigonometric

$^3$ According to certain indications (fairly obscure) from Lipschitz [205 a], Dirichlet might have believed that if the set of points of discontinuity was sparse, the "derivative" is finite, and would have in all cases limited his investigations to the case where this is so.

$^4$ From Dirichlet and Riemann to our days, we will see this close association between integration and what we now call "harmonic analysis" being pursued, which provides in some way a touch stone.

series has not been studied by Dirichlet, that are of importance" ([259 a], pp. 237-238).

The idea of Riemann is to start from the approximation procedure for the integral, brought back to respectability by Cauchy, and to determine when the "Riemann sums" of a function $f$, in a bounded interval $[a, b]$, tend to a limit (the maximal length of the intervals in the subdivision tending to 0); a problem of which he obtains without trouble the solution in the following form: for every $\alpha > 0$, there is a subdivision of $[a, b]$ into partial intervals of sufficiently small maximal length so that the sum of the lengths of the intervals of this subdivision, where the oscillation of $f$ is $> \alpha$ is arbitrarily small. He shows as well that this condition is satisfied, not only for continuous and piecewise monotonic functions, but also for functions that may have an everywhere dense set of points of discontinuity.$^5$

The memoir of Riemann was not published until after his death, in 1867. But this time, the times were more favourable for this kind of research, and the "Riemann integral" took its place naturally in the stream of ideas that was leading then to a sustained study of the "continuum" and of functions of real variables (Weierstrass, Du Bois-Reymond, Hankel, Dini) and was to end up, with Cantor, at the blossoming of set theory. The form given by Riemann to the integrability condition suggested the idea of "measure" for the set of points of discontinuity of a function in an interval; but nearly 30 years were to pass by before a fruitful and convenient definition of this notion would be given.

The first attempts in this direction are due to Stolz, Harnack and Cantor (1884-1885); the first two, in order to define the "measure" of a bounded subset $E$ of $\mathbf{R}$, consider sets $F \supset E$ which are finite unions of intervals, take for each $F$ the sum of the lengths of the corresponding intervals, and say that the "measure" of $E$ is the lower bound of these numbers; whereas Cantor, starting from the beginning in the space $\mathbf{R}^n$, considers for a bounded set $E$ and for $\rho > 0$ the neighbourhood $V(\rho)$ of $E$ constructed of points whose distance to $E$ is $\leq \rho$, and takes the lower bound of the "volume" of $V(\rho)$.$^6$ With this definition, the "measure" of a set is equal to that of its closure, whence in particular it is possible that the "measure" of the union of two sets with no point in common can be strictly less that the sum of the "measures" of these two sets. No doubt in order to alleviate this last difficulty, Peano [246 a] and Jordan ([174 c], v. I, pp. 28-31), a few years later, introduce, alongside the "measure" of Cantor $\mu(A)$ of a set $A$ contained in a block $I$, its "interior measure" $\mu(I) - \mu(I - A)$, and call "measurable" the sets $A$ (now

$^5$ On the other hand, H. J. Smith gave, already in 1875, the first example of a function which is not integrable in the sense of Riemann, and whose set of points of discontinuity is sparse ([287], v. II, pp. 86-100).

$^6$ Cantor does not give a precise definition of this "volume" and limits himself to saying that it can be calculated by a multiple integral ([47], pp. 229-236 and 257-258). It is easy to see that his definition is equivalent to that of Stolz-Harnack, by an application of the Borel-Lebesgue theorem.

called "quarriable") for which these two numbers coincide. The union of two quarriable sets $A, B$ with no point in common is then quarriable and has as "measure" the sum of the "measures" of $A$ and $B$; but an open bounded set is not necessarily quarriable, and neither is the set of rational numbers contained in a bounded interval, which removed much of the interest in the notion of Peano-Jordan.

It is to E. Borel [32 a] that the credit for having known how to discern the faults of the previous definitions and for having seen how it was possible to remedy them is due. It was known since Cantor that every open set $U$ in $\mathbf{R}$ is the union of a countable family of its "components", pairwise disjoint open intervals; instead of seeking to approach $U$ "from the outside" by enclosing it in a finite sequence of intervals, Borel, relying on the preceding result, proposes taking as the measure of $U$ (when $U$ is bounded) the sum of the lengths of its components. Then he describes very summarily$^7$ the class of sets (also called "Borel") that can be obtained, starting from the open sets, by iterating indefinitely the operations of countable union and of "difference" $A - B$, and indicates that, for these sets, a measure can be defined that possesses the fundamental property of complete additivity: if a sequence $(A_n)$ is made up of pairwise disjoint Borel sets, the measure of their union (assumed bounded) is equal to the sum of their measures.

This definition was to inaugurate a new era in Analysis: on the one hand, in tandem with the contemporary work of Baire, it became the point of departure for a whole series of researches of a topological nature on the classification of sets of points (see p. 165); and above all, it was to be of use as the basis of the extension of the notion of integral, created by Lebesgue in the first years of the XXth century.

In his thesis [196 a], Lebesgue starts by making precise and developing the succinct indications of E. Borel; imitating the method of Peano-Jordan, the "exterior measure" of a bounded set $A \subset \mathbf{R}$ is defined as the lower bound of the measures of the open sets containing $A$; then, if $I$ is an interval containing $A$, the "interior measure" of $A$ is the difference between the exterior measures of $I$ and of $I - A$; thus is obtained a notion of a "measurable set" which only differs from the initial "constructive" definition of Borel by the adjunction of a subset of a set of measure zero in the sense of Borel. This definition extended immediately to the spaces $\mathbf{R}^n$; the old conception of the definite integral $\int_a^b f(t) dt$ of a bounded function $\geq 0$ as the "area" bounded by the curve $y = f(x)$, the straight lines $x = a, x = b$ and $y = 0$, supplied thus an immediate extension of the Riemann integral to all the functions $f$ for which the measure of the preceding set was defined. But the originality of

$^7$ Measure is still not for Borel, at that time, anything other than a technical means for the study of certain series of rational functions, and he underlines himself that, for the goal that he proposes for himself, the usefulness of measure is due above all to the fact that a set of non zero measure is not countable ([32 a], p. 48).

Lebesgue does not reside so much in the idea of this extension,$^8$ than in his discovery of the fundamental theorem on the passage to the limit in the integral thus conceived, a theorem which appears in his work as a consequence of the complete additivity of the measure;$^9$ he realises immediately all its importance, and makes it the key stone of the didactic exposé of his theory that he gives, already in 1904, in the famous "Lessons on integration and the quest for primitive functions" [196 c].$^{10}$

We cannot describe here in detail the innumerable advances that the results of Lebesgue were to bring about in the study of the classical problems of infinitesimal Calculus. Lebesgue himself had already, in his thesis, applied his theory to the extension of the classical notions of length and area to more general sets than the usual curves and surfaces; concerning the considerable development of this theory over half a century, we send the reader back to the recent exposé by L. Cesari [59]. Let us mention also the applications to trigonometric series, developed by Lebesgue almost immediately after his thesis [196 b], and which were to open new horizons for this theory, whose exploration is far from being over in our day (see [345]). Finally and especially, the definition of the spaces $L^p$ and the Fischer-Riesz theorem ([111], [260 a and c]; cf. p. 213) brought to light the role that could be played in Functional Analysis by the new notion of integral; a role that could only grow with the subsequent generalisations of this notion, of which we shall speak in a moment.

Beforehand, we will stop a little longer with one of the problems to which Lebesgue devoted the most effort, the link between the notions of integral and of primitive. With the generalisation of the integral introduced by Riemann the question that arose naturally was to know if the classical correspondence between integral and primitive, valid for continuous functions, still subsisted in the more general case. But it is easy to give examples of functions $f$, integrable in the Riemann sense, and such that $\int_a^x f(t)dt$ does not have a derivative (nor even a derivative on the right or on the left) at certain points;

$^8$ Independently of Lebesgue, W. H. Young had had the same idea for semi-continuous functions [339 a].

$^9$ The particular case of this theorem, where we have the case of a sequence of functions integrable in a compact interval in the Riemann sense, uniformly bounded, and whose limit is integrable in the Riemann sense, had been proved by Arzelà [9 a].

$^{10}$ Among the most important consequences of this theorem in the general theory of integration, Egoroff's theorem on the convergence of sequences of measurable functions [99] must be mentioned in particular, making precise earlier remarks of Borel and Lebesgue. On the other hand, measurable (numerical) functions had first been defined by Lebesgue by the property that, for such a function $f$, the reciprocal image by $f$ of every interval of $\mathbf{R}$ is a measurable set. But from 1903, Borel and Lebesgue had drawn attention to the topological properties of these functions; they were put in their definitive form by Vitali, who, in 1905 [320 a], was the first to formulate the property of measurable functions known ordinarily under the name of "Lusin's theorem" (who found it again in 1912).

conversely, Volterra had shown, in 1881, that a function $F(x)$ may have a bounded derivative in an interval $I$, but be non integrable (in the Riemann sense) in $I$. By an analysis of great subtlety (where the theorem on the passage to the limit in the integral is far from being sufficient), Lebesgue succeeds in showing that, if $f$ is integrable (in his sense) in $[a, b]$, $F(x) = \int_a^x f(t)dt$ has almost everywhere a derivative equal to $f(x)$ [196 c]. Conversely, if a function $g$ is differentiable in $[a, b]$ and if its derivative $g' = f$ is bounded, $f$ is integrable and we have the formula $g(x) - g(a) = \int_a^x f(t)dt$. But Lebesgue notes that the problem is much more complex when $g'$ is not bounded; $g'$ is not necessarily integrable in this case, and the first problem is therefore to characterise the continuous functions $g$ for which $g'$ exists almost everywhere and is integrable. By limiting himself to the case where one of the derived numbers$^{11}$ of $g$ is everywhere finite, Lebesgue showed that $g$ is necessarily a function with bounded variation.$^{12}$ Finally he establishes the converse of this last result: a function with bounded variation $g$ has almost everywhere a derivative, and $g'$ is integrable; but it is no longer necessarily true that

the difference between the two parts of this relation is a function of bounded variation which is not constant and with zero derivative almost everywhere (a "singular" function). It remained to characterise the functions with bounded variation $g$ such that the relation (22.2) held. Lebesgue established that these functions (called "absolutely continuous" by Vitali, who made a detailed study of them) are those that have the following property: the total variation of $g$ in an open set $U$ (sum of the total variations of $g$ in each of the connected components of $U$) tends to 0 with the measure of $U$.

We will see below how, in a weakened form, these results would later acquire a much more general scope. In their initial form, their field of application remained fairly restricted, and did not go beyond the framework of the "fine" theory of functions of a real variable, of which we will not study here the subsequent development; we will be content with mentioning the deep work of Denjoy and his followers and successors (Perron, de la Vallée-Poussin, Khintchine, Lusin, Banach, etc. ); the reader will find a detailed exposé in the book of S. Saks [268].

$^{11}$ The derived numbers on the right of $g$ at a point $x$ are the two limits $\lim_{h \to 0} \sup_{h > 0} (g(x + h) - g(x))/h, \lim_{h \to 0} \inf_{h > 0} (g(x + h) - g(x))/h$. The derived numbers on the left are defined similarly.

$^{12}$ These functions had been introduced by Jordan, in connection with the rectification of curves [174 c]; he showed that one can give two equivalent definitions as follows: a) $f$ is the difference between two increasing functions; b) for every subdivision of the interval $[a, b]$ by a finite sequence of increasing points $(x_i)_{0 \leq i \leq n}$, with $a = x_0, b = x_n$, the sum $\sum_{i=1}^n |f(x_i) - f(x_{i-1})|$ is bounded by a number independent of the subdivision under consideration. The upper bound of these sums is the total variation of $f$ in $[a, b]$.

One of the essential advances brought about by the Lebesgue theory concerns multiple integrals. This notion was introduced about the middle of the XVIIIth century, and first in the form of an “indefinite integral” (by analogy with the theory of the integral of functions of a single variable, $\int \int f(x, y) dx dy$ denotes a solution of the equation $\frac{\partial^2 z}{\partial x \partial y} = f(x, y)$); but already from 1770, Euler had a very clear conception of the double integral extended to a bounded domain (bounded by arcs of analytic curves), and wrote down correctly the formula for evaluating such an integral by means of two simple integrals in succession ([108 a], (1), v. XVII, pp. 289–315). It was not difficult to justify this formula starting from “Riemann sums”, so long as the function being integrated was continuous, and the domain of integration not too complicated; but as soon as one wanted to tackle more general cases, the procedure of Riemann ran into serious difficulties ($f(x, y)$ may be integrable in the Riemann sense, without $\int dx \int f(x, y) dy$ having a meaning when the simple integrals are taken in the Riemann sense). These difficulties vanish when we pass to the definition of Lebesgue; already this latter had showed in his thesis that, when $f(x, y)$ is a bounded “Baire function”, it is the same for the functions $y \mapsto f(x, y)$ (for all $x$) and $x \mapsto \int f(x, y) dy$, and we have the formula

(integral taken in the rectangle). A little later, Fubini [121] brought an important complement to this result by proving that, if it is only assumed that $f$ is integrable, then the set of $x$ such that $y \mapsto f(x, y)$ is not integrable is of zero measure, which allowed the extension of the formula (22.3) to this case.

Finally, in 1910 [157 e], Lebesgue tackles the extension to multiple integrals of his results on the derivatives of simple integrals. He is thus brought to associate with a function $f$, integrable on all compact subsets of $\mathbf{R}^n$, the function of a set $F(E) = \int_E f(x) dx$, defined for all integrable subsets $E$ of $\mathbf{R}^n$, which generalises the concept of “indefinite integral”; and he observes on this occasion that this function possesses the following two properties: $1^\circ$ it is completely additive; $2^\circ$ it is “absolutely continuous” in the sense that $F(E)$ tends to 0 with the measure of $E$. The essential part of the memoir of Lebesgue consists in proving the converse of this proposition.$^{13}$ But he does not stop there, and in the same direction, signals the possibility of generalising the notion of function with bounded variation, by considering functions of a measurable set $F(E)$, completely additive and such that $\sum_n |F(E_n)|$ remains bounded for every countable partition of $E$ into measurable subsets $E_n$. And, if he limits himself in fact to not considering such functions except on the set of blocks of $\mathbf{R}^n$, it is quite clear that there only remained one step to take to end up with the general notion of measure that J. Radon is going

$^{13}$ The main tool, in this proof, is a covering theorem, proved sometime before by Vitali [320 b], and which has remained fundamental in this type of question.

to define in 1913, gathering into the same synthesis the Lebesgue integral and the Stieltjes integral, of which we must now speak.

In 1894, T. Stieltjes published, under the title "Research on continuous fractions" [297], a very original memoir where, starting from a question which is very restricted in appearance, were to be found stated and solved, with a rare elegance, problems of a quite new nature in the theory of analytic functions and that of functions of a real variable.$^{14}$ In order to represent the limit of a certain sequence of analytic functions, Stieltjes had been brought, among others, to introduce on the straight line, the concept of a positive "mass distribution", a familiar notion for a long time in the physical sciences, but which until then had never been considered in mathematics except under restrictive hypotheses (in general, the existence of a "density" at every point, varying in a continuous way); he remarks that giving such a distribution is equivalent to that of an increasing function $\phi(x)$ which gives the total mass contained in the interval with ends 0 and $x$ for $x > 0$, and this mass having changed sign for $x < 0$, the discontinuities of $\phi$ corresponding to masses "concentrated at a point".$^{15}$ Stieltjes then forms, for such a mass distribution in an interval $[a, b]$, the "Riemann sums" $\sum_i f(\xi_i)(\phi(x_{i+1}) - \phi(x_i))$ and shows that, when $f$ is continuous in $[a, b]$, these sums tend to a limit that he denotes $\int_a^b f(x)d\phi(x)$. Only needing to integrate continuous functions (and even differentiable functions) Stieltjes did not push further forward with the study of this integral$^{16}$ and for about ten years this notion does not seem to have attracted attention.$^{17}$ But in 1909, F. Riesz [260 d], solving a problem stated a few years earlier by Hadamard (cf. p. 213), proved that the Stieltjes integrals $f \mapsto \int_a^b fd\phi$ are the most general continuous linear functionals on the space $C(I)$ of numerical functions continuous on $I = [a, b]$ ($C(I)$ being furnished with the uniform convergence topology$^{18}$); and the elegance and simplicity of this result caused almost immediately several generalisations of

$^{14}$ It is there among others that is formulated and solved the famous "problem of moments" (cf. p. 215). $^{15}$ Stieltjes still does not make any distinction between the various types of intervals having the same ends $a, b$, which leads him to conceive that at the points of discontinuity $c$ of $\phi$, part of the mass concentrated at $c$ belongs to the interval of origin $c$, and the other part to the interval of end $c$, according to the value of $\phi(c)$. $^{16}$ The first appearance must however be noted here, with Stieltjes, of the idea of "convergence" of a sequence of measures ([297], p. 95; it is a case in fact of the strong limit). $^{17}$ It acquires however importance with the development by Hilbert and his school, of the spectral theory of operators, starting in 1906. It is on this occasion that Hellinger, around 1907, defined integrals such as that which he denotes $\int \frac{(dg)^2}{df}$, and which appear at first more general than those of Stieltjes; but in fact, Hahn showed, already in 1912, that they reduce to this latter. $^{18}$ It is also in this work that the notion appears of the weak limit of a sequence of measures ([260 d], p. 49).

it. The most successful was that of J. Radon, in 1913 [256]: combining the ideas of F. Riesz and of Lebesgue, he showed how an integral could be defined by the procedures of Lebesgue, starting with an arbitrary “completely additive function of a set” (defined on measurable sets for the Lebesgue measure) instead of starting with the Lebesgue measure. In the notion of “Radon measure” on $\mathbf{R}^n$, thus defined, that of function with “bounded variation” is found to be absorbed: the decomposition of such a function as a difference of two increasing functions is a particular case of the decomposition of a measure as a difference of two positive measures; in the same way, the “measure with base $\mu$” corresponds to the notion of an “absolutely continuous” function, and the decomposition of an arbitrary measure into a measure with base $\mu$ and a measure foreign to $\mu$, to the Lebesgue decomposition of a function with bounded variation as the sum of an absolutely continuous function and a “singular” function. As well, Radon showed that the “density” with respect to $\mu$ of a measure with base $\mu$ exists still when $\mu$ is a measure having as base the Lebesgue measure, by using a previous idea of F. Riesz (taken up again and popularised later by J. von Neumann among others), which consists in constructing an image of the measure $\mu$ by a map $\theta$ of $\mathbf{R}^n$ into $\mathbf{R}$, chosen so that $\theta(\mu)$ is the Lebesgue measure on $\mathbf{R}$.

Almost immediately after the appearance of the memoir of Radon, Fréchet remarked that nearly all the results of this work could be extended to the case where the “completely additive functions of a set”, instead of being defined on the measurable sets of $\mathbf{R}^n$, are defined for certain subsets of an arbitrary set $E$ (these subsets being such that the operations of countable union and of “difference” still give sets for which the function is defined). However, the expression of a measure with base $\mu$ in the form $g.\mu$ relied, with Lebesgue and Radon, on arguments using in an essential way the topology of $\mathbf{R}^n$ (and we have seen that the proof of Radon only applies if $\mu$ is a measure with base the Lebesgue measure); it is only in 1930 that O. Nikodym [235] obtained this theorem in its general form, by a direct argument (notably simplified a few years later by J. von Neumann, thanks to the use of the properties of the spaces $L^2$ ([324 g], pp. 127-130)).

With the memoir of Radon, the general theory of integration could be considered as completed in its main lines; as subsequent substantial acquisitions, it is only possible to mention the definition of the infinite product of measures, due to Daniell [76 b], and that of the integral of a function with values in a Banach space, given by Bochner in 1933 [24 a], and which was a prelude to the more general study of the “weak integral” developed a few years later by Gelfand [125 a], Dunford and Pettis ([95] and [96]). But it remained to popularise the new theory, and to make of it a mathematical instrument in general use, while the majority of mathematicians, around 1910, still saw in the “Lebesgue integral” only an instrument of great precision, delicate handling, destined only for researches of extreme subtlety and of an extreme abstraction. That was the work of Carathéodory, in a book which remained a classic for a long time [49] and which enriched Radon’s theory with numerous original remarks.

But it is with this book also that the notion of integral, which had been in the front line of the concerns of Lebesgue (as was sufficiently shown by the titles of his thesis [196 a] and of his main work on these questions [196 c]) yields the field for the first time to that of measure, which had been for Lebesgue (as before him with Jordan) an auxiliary technical means. This change in point of view was due no doubt, with Carathéodory, to the excessive importance that he seems to have attached to “p-dimensional measures”.$^{19}$ Since then, the authors that have treated integration are divided between these two points of view, not without involving themselves in debates which have caused a lot of ink to flow if not a lot of blood. One group have followed Carathéodory; in their exposés ever more abstract and more axiomatised, measure, with all the technical refinements to which it lends itself, not only plays the dominant role, but also tends to lose contact with the topological structures to which it is in fact linked in the majority of problems where it is involved. Other exposés follow more or less closely a method already indicated in 1911 by W. H. Young, in a memoir unfortunately little noticed [339 b], and developed since by Daniell. The first, dealing with the Lebesgue integral, parted company with the “Cauchy integral” of continuous functions with compact support, assumed known, in order to define successively the upper integral of lower semi-continuous functions, then of arbitrary numerical functions, whence a definition of integrable functions, modelled on that of Lebesgue for sets, by purely “functional” means. Daniell, in 1918 ([76 a]; cf. [207]) extended this exposé, with some variants, to functions defined on an arbitrary set; in the same stream of ideas (and in close link with the methods used in spectral theory before Gelfand), we must also point out the memoir of F. Riesz [260 h] which puts into an elegant and concise form the few results of the theory of ordered spaces that play a role in the theory of Integration.

Rather than in works of exposition, more or less pleasant to read, but whose substantial content could not vary much, it is on the side of applications that progress arising from the theory of Integration since 1920 must be sought: the theory of probability (at other times the pretext for guessing and paradoxes, and become a branch of the theory of Integration since its

$^{19}$ It is a case there of the generalisation of the “length of a plane curve” to arbitrary values n and p of the dimension of the ambient space and of the dimension of the space being studied; it is assumed of course that we have $0 \leq p \leq n$, but it is not always assumed that p is an integer. This question has been the object of the work of numerous authors since Minkowski, Carathéodory and Hausdorff; Lebesgue himself, who tackles some particular cases in his thesis, seems not to have seen there other than an opportunity to put to the test the power of the tools that he had just forged.

axiomatisation by Kolmogoroff [183], but an autonomous branch with its own methods and problems); ergodic theory; spectral theory and harmonic analysis, since the discovery by Haar of the measure that bears his name, and the movement of ideas provoked by this discovery, have made of the integral one of the most important tools in the theory of groups.