23. Haar measure. Convolution.

The notions of length, area and volume among the Greeks are essentially based on their invariance under displacements: "Things which coincide ($\varepsilon \phi \alpha \rho \mu o \xi o \nu \tau \alpha$) are equal" (Eucl. El., Book I, "Common notions" 4); and it is by an ingenious use of this principle that all the formulae are obtained giving the areas or volumes of the classical "figures" (polygons, conics, polyhedra, spheres, etc.), sometimes by a procedure of finite decomposition, sometimes by "exhaustion".$^1$ In modern language, it can be said that what was done by the Greek geometers, is to prove the existence of "functions of sets" which are additive and invariant under displacements, but defined only for sets of a very particular type. The integral calculus can be considered as replying to the need for enlarging the domain of definition of these functions of sets, and, from Cavalieri to H. Lebesgue, it is this preoccupation that will be in the forefront of the research of the analysts; as for the property of invariance under displacements, it becomes a secondary consideration, having become a trivial consequence of the general formula for change of variables in double or triple integrals and because of the fact that an orthogonal transformation has determinant equal to $\pm 1$. Even in the non-Euclidean geometries (where however the group of displacements is different), the point of view remains the same: in a general way, Riemann defines the infinitesimal elements of area or volume (or their analogues for dimensions $\geq 3$) starting with a $ds^2$ by means of the classical Euclidean formulae, and their invariance under transformations that leave invariant the $ds^2$ is thus almost a tautology.

It is only around 1890 that other less immediate extensions of the notion of measure invariant under a group appear, with the development of the theory of the integral invariants, notably by H. Poincaré and E. Cartan;

$^1$ It can be shown that if two polygons $P, P'$ have the same area, there are two polygons $R \supset P, R' \supset P'$ which can each be decomposed into a finite number of polygons $R_i$ (resp. $R'_i$) ($1 \leq i \leq m$) without any interior point in common, such that $R_i$ and $R'_i$ can be obtained one from the other by a displacement (depending on $i$), and such that $R$ (resp. $R'$) are the union of a finite family of polygons $S_j$ (resp. $S'_j$) ($0 \leq j \leq n$) without any interior points in common, with $S_0 = P, S'_0 = P'$, and $S'_j$ being obtained from $S_j$ by a displacement for $1 \leq j \leq n$. On the other hand, M. Dehn has proved [81] that this property is not valid for the volume of polyhedra, and that the procedures of exhaustion used since Eudoxus were in consequence inevitable.

H. Poincaré only foresees groups with one parameter operating on a part of space, whereas E. Cartan is mainly interested in groups of displacements, but operating in other spaces than those in which they are defined. For example, he determines thus among others ([52 a], v. II₁, pp. 265-302) the invariant (under a group of displacements) measure in the space of straight lines in $\mathbf{R}^2$ or in $\mathbf{R}^3$;$^2$ further, he points out that in general the integral invariants for a Lie group are none other than particular differential invariants, and that it is thus possible to determine all of them by the methods of Lie. It seems however that no one thought to consider or to use a measure invariant on the group itself before the fundamental work of A. Hurwitz in 1897 [168]. Seeking to construct polynomials (on $\mathbf{R}^n$) invariant under the orthogonal group, Hurwitz starts from the remark that, for a finite group of linear transformations, the problem is solved immediately by taking the mean of the transforms $s.P$ of an arbitrary polynomial $P$ by all the elements $s$ of the group; which gives him the idea of replacing, for the orthogonal group, the mean by an integral relative to an invariant measure; he gives explicitly the expression for this latter by means of the parametric representation by the angles of Euler, but observes immediately (independently of E. Cartan) that the methods of Lie supply the existence of an invariant measure for all Lie groups. Perhaps because of the decline of the theory of invariants at the beginning of the XXth century, the ideas of Hurwitz had hardly any immediate reaction, and would not be considered valuable until after 1924, with the extension to compact Lie groups, by I. Schur and H. Weyl, of the classical theory of Frobenius (p. 121) on the linear representations of finite groups. The first limits himself to the case of the orthogonal group, and shows how the method of Hurwitz allows the extension of the classical relations of orthogonality of characters; an idea that H. Weyl combines with the work of E. Cartan on semisimple Lie algebras to obtain the explicit expression for the characters of the irreducible representations of compact Lie groups and the complete reducibility theorem [331 b], then, by a bold extension of the notion of "regular representation", the famous theorem of Peter-Weyl, a perfect analogy of the decomposition of the regular representation into its irreducible components in the theory of finite groups [332].

One year previously, O. Schreier had founded the general theory of topological groups [276 a], and from that moment it was clear that the arguments of the memoir of Peter-Weyl would remain valid as such for every topological group on which could be defined an "invariant measure". Truth to tell, the general notions of topology and measure were still at this time in full formation, and neither the category of topological groups on which one could hope to define an invariant measure, nor the sets for which this "measure" would be defined, seemed clearly delimited. The only evident point was that one

$^2$ The invariant measure on the space of straight lines in the plane had already been essentially determined in connection with the problems of "geometric probability", notably by Crofton, whose work E. Cartan probably did not know at that time.

could not hope to extend to the general case the infinitesimal methods proving the existence of an invariant measure on a Lie group. But, another stream of ideas, issuing from work on the Lebesgue measure, was leading precisely to more direct methods of attack. Hausdorff had proved, in 1914, that there does not exist an additive function on sets not identically zero defined for all subsets of $\mathbf{R}^3$ and invariant under displacements, and it was natural to find out if this result was still valid for $\mathbf{R}$ and $\mathbf{R}^2$: a problem that was solved by S. Banach in 1923 in a surprising way, by showing on the contrary that in these two cases such a “measure” well and truly existed [14 b]; his procedure, very ingenious, relied already on a construction by transfinite induction and on the consideration of “means” $\frac{1}{n} \sum_{k=1}^n f(x + \alpha_k)$ of translates of a function by elements of the group.$^3$ It is analogous ideas that allowed A. Haar, in 1933 [139], to make the decisive step by proving the existence of an invariant measure for locally compact groups with a countable base of open sets: inspired by the method for the approximation of a volume, in classical integral Calculus, by a juxtaposition of congruent cubes with arbitrarily small sides, he obtains, with the help of the diagonal procedure, the invariant measure as the limit of a sequence of “approximating measures”, a procedure that still is essentially the one used today. This discovery had a very big repercussion, in particular because it allowed J. von Neumann to solve immediately, for compact groups, Hilbert’s famous “5th problem” on the characterisation of Lie groups by purely topological properties (to the exclusion of all previously given differential structure). But it was noticed immediately that, in order to be able to use the invariant measure in an efficacious way, it was not enough only to know of its existence, but also to know that it was unique up to a factor; this point was first proved by J. von Neumann for compact groups, by using a method for the definition of the Haar measure by “means” of continuous functions, analogous to those of Banach [324 e]; then J. von Neumann [324 f] and A. Weil [330 c], by different methods, obtained simultaneously the uniqueness in the case of locally compact groups, A. Weil indicating at the same time how the procedure of Haar could be extended to general locally compact groups. It is also A. Weil (loc. cit.) who obtained the condition for the existence of a relatively invariant measure on a homogeneous space, and showed finally that the existence of a “measure” (endowed with reasonable properties) on a separated topological group, implied ipso facto that the group is locally precompact. These works completed essentially the general theory of Haar measure; the only more recent addition that needs quoting is the notion of quasi-invariant measure, which hardly emerged until around 1950, linked with the theory of representations of locally compact groups in Hilbert spaces.

$^3$ J. von Neumann showed, in 1929, that the deep reason for the difference in behaviour between $\mathbf{R}$ and $\mathbf{R}^2$ on the one hand, and the $\mathbf{R}^n$ for $n \geq 3$ on the other, must be sought for in the commutativity of the group of rotations in the space $\mathbf{R}^2$.

The history of the convolution product is more complex. Already from the beginning of the XIXth century, it is noticed that if, for example, $F(x, t)$ is an integral of a partial differential equation in $x$ and $t$, linear and with constant coefficients, then

is also an integral of the same equation; Poisson, among others, already before 1820, uses this idea to write the integrals of the heat equation in the form

A little later, the expression

for the partial sum of a Fourier series and the study, made by Dirichlet, of the limit of this integral when $n$ tends to $+\infty$, gives the first example of "regularisation" $f \mapsto \rho_n * f$ on the torus $T$ (in actual fact by a sequence of non-positive "kernels" $\rho_n$, which complicates considerably its study); under the name of "singular integrals", the analogous integral expressions will be a favourite subject for the analysts of the end of the XIXth century and the beginning of the XXth century, from P. du Bois-Reymond to H. Lebesgue. On $\mathbf{R}$, Weierstrass uses the integral (23.1) for a proof of his theorem on approximation by polynomials, and gives in this connection the general principle of regularisation by a sequence of positive "kernels" $\rho_n$ of the form $x \mapsto c_n \rho(nx)$. On $T$, the most famous example of a regularisation by positive kernels is given a little later by Fejér, and from that moment, it is the standard procedure which will be at the basis of the majority of "summation methods" of series of functions.

However, this work, because of the lack of symmetry of the roles that are played by the "kernel" and the regularised function, hardly made apparent the algebraic properties of the convolution product. It is to Volterra especially that is owed the stressing of this point. He studies in a general way the "composition" $F * G$ of two functions of two variables

that he sees as a generalisation, by "passing from the finite to the infinite", of the product of two matrices [322 a]. He distinguishes very early on the case ("of the closed cycle" so called because of its interpretation in the theory of heredity) where $F$ and $G$ depend only on $y - x$; it is then the same with $H = F * G$, and if we set $F(x, y) = f(y - x), G(x, y) = g(y - x)$, we have

Meanwhile, the algebraic formalism developed by Volterra did not bring out the links with the group structure of $\mathbf{R}$ and the Fourier transform. We do not need to write the history of this latter here; but it is appropriate to note that starting with Cauchy the analysts who deal with the Fourier integral are mainly concerned with finding wider and wider conditions for the validity of the different formulae of "inversion", and neglect somewhat its algebraic properties. It is certainly not possible to say as much of the work of Fourier himself (or of that of Laplace on the analogous integral $\int_0^{+\infty} e^{-st}f(t)dt$); but these transformations had been introduced essentially in connection with linear problems, and it is therefore not very surprising that for a long time it had not been thought worth considering the product of two Fourier transforms (with the exception of the products of trigonometric series or of entire series, but the link with the convolution of discrete measures could obviously not have been noticed in the XIXth century). The first mention of this product and of the convolution on $\mathbf{R}$ is probably to be found in a memoir of Tchebychef [306], in connection with questions about the calculation of probabilities. Indeed, in this theory, the convolution $\mu * \nu$ of two "probability laws" on $\mathbf{R}$ (positive measures of total mass 1) is none other than the "product" probability law of $\mu$ and $\nu$ (for the sum of the corresponding "random variables"). Of course, with Tchebychef, it is still only a matter of the convolution of probability laws having a density (with respect to the Lebesgue measure), hence of the convolution of functions; besides it only occurs with him in an episodic way, and it will be thus in the few rare works where it appears before the period 1920-1930. In 1920, P.-J. Daniell, in a note rarely consulted [76 c], defines the convolution of two arbitrary measures on $\mathbf{R}$ and the Fourier transform of such a measure, and observes explicitly that the Fourier transform takes the convolution to the ordinary product; a formalism that, starting in 1925, is going to be intensively used by the probabilists, following P. Lévy especially. But the fundamental importance of the convolution in the theory of groups is only fully recognised by H. Weyl in 1927; he realises that, for a compact group, the convolution of functions plays the role of multiplication in the group algebra of a finite group, and allows him as a consequence to define the "regular representation"; at the same time, he finds in the regularisation the equivalent of the unit element of the algebra of a finite group. It remained to make the synthesis of all these points of view, which is accomplished in the book of A. Weil [330 c], a prelude to subsequent generalisations that will constitute, on the one hand the normed algebras of I. Gelfand, and on the other the convolution of distributions.

The Haar measure and convolution have rapidly become essential tools in the tendency to algebraisation that distinguishes so strongly modern Analysis; we will have to mention numerous applications of it in subsequent notes. We will limit ourselves here to pointing out that one which concerns the "variation" of closed subgroups (and notably discrete subgroups) of a locally compact group. This theory, starting with a result of K. Mahler in the Geometry of numbers, was inaugurated in 1950 by C. Chabauty, and has just been considerably developed and deepened by Macbeath and Swierczkowski [212].