Introduction

The concept of measure of magnitudes is fundamental, as well in everyday life (length, area, volume, weight) as in experimental science (electric charge, magnetic mass, etc.). The common characteristic of the 'measures' of such diverse magnitudes lies in the association of a number to each portion of space fulfilling certain conditions, in such a way that, to the union of two such portions (assumed to be without common point), there corresponds the sum of the numbers assigned to each of them (the additivity of the measure) (). Moreover, the measure is usually a positive number, and this implies that it is an increasing function of the portion of space measured (*). It will be observed on the other hand that in practice, one hardly ever worries about specifying which portions of space are to be regarded as 'measurable'; it is of course indispensable to settle this matter unambiguously in every mathematical theory of measure; for example, this is what one does in elementary geometry when one defines the area of polygons or the volume of polyhedra; in all of these cases, the family of 'measurable' sets must naturally be such that the union of any two of them having no point in common is also 'measurable'.

In most of the above examples, the measure of a portion of space tends to 0 with its diameter: classically, a point 'has no length', which means that it is contained in intervals of arbitrarily small length, consequently one can only assign to it the length 0; the measures of such magnitudes are said to be 'diffuse'. However, developments in Mechanics and Physics have introduced the notion of magnitudes for which an object of negligible dimensions may

(*) It is not obvious $a$ priori that different species of magnitudes can be measured by the same numbers, and it is undoubtedly by deepening the concept of the measure of magnitudes that the Greeks arrived at their theory of ratios of magnitudes, equivalent to that of the real numbers $> 0$ (cf. GT, Ch. V, §2 and the Historical Note of Ch. IV).

(**) This does not apply, for example, to the electric charge of a body; however, the measure of the total electric charge may be regarded as the difference of the measure of the positive electric charges and the measure of the negative electric charges, both of which are positive measures.

still have non-negligible measure: gravitational or electrical 'point masses', which, to tell the truth, are largely mathematical fictions more than they are strictly experimental notions. One is thus led, in Mathematics, to consider measures defined as follows: to each point $a_i$ ($1 \leq i \leq n$) of a finite set F there is attached a number $m_i$, its 'mass' or its 'weight', and the measure of an arbitrary set A is the sum of the masses $m_i$ of the points $a_i$ that belong to A.

Closely tied to the concept of measure is that of weighted sum. For example, consider in space a finite number of masses (gravitational or electrical) $m_i$ placed at points $a_i$ (with coordinates $x_i, y_i, z_i$); the component along Oz (for example) of the attraction exerted on a point b (with mass 1 and coordinates $\alpha, \beta, \gamma$) by the set of these masses is (for a suitable system of units) the sum

$r_i^2 = (x_i - \alpha)^2 + (y_i - \beta)^2 + (z_i - \gamma)^2$ being the square of the distance between the points $a_i$ and b. In other words, one considers the value of the function

at each point $a_i$, one multiplies it by the 'weight' of this point, and one sums up the 'weighted values' of $f$ so obtained. It is known that such sums intervene constantly in Mechanics: centers of gravity and moments of inertia are the best-known examples.

If one wants to extend the notion of 'weighted sum' from the case of point masses to that of a 'diffuse' measure, where every point has measure zero, one finds oneself in the presence of the problem, of so paradoxical an aspect, that gave rise to the Integral Calculus: how to assign a meaning to a 'sum' with infinitely many terms each of which, taken by itself, is zero. Let us take up again the example of calculating the attraction exerted on a point, when the attracting masses are 'distributed continuously' throughout a volume V. If V is decomposed into a finite number of (pairwise disjoint) subsets $V_i$, one assumes that the component along Oz of the attraction exerted by V on a point b is the sum of the components of the attractions exerted on b by each of the $V_i$. But if the diameter of each $V_i$ is small, the continuous function $f(x, y, z)$ varies little in $V_i$, and one is led to liken the attraction exerted by $V_i$ to that which would be exerted by a point mass equal to the mass $m_i$ of $V_i$ and placed at any point $a_i$ of the volume $V_i$. One is thus led to take, as an approximate value of the sought-for number, the 'Riemann sum' $\sum_i m_i f(x_i, y_i, z_i)$; for this to be justified from the mathematical point of view, it must naturally be proved that these approximate values tend to a limit as the maximum diameter of the $V_i$ tends to 0, which is an easy consequence of the uniform continuity of the function $f$ in $V$ (assuming $V$ is compact and the point $b$ is not in $V$).

It is known that the 'method of exhaustion' of the Greeks and 'Cavalieri's principle' for the systematic calculation of plane areas and of volumes are based on an analogous procedure, by the decomposition into 'slices' of the areas and volumes considered; the 'weighted sums' thus arrived at are none other than the integrals $\int_a^b f(x)\, dx$ (cf. the Historical Note for Chs. I–III of Book IV). Here again, it is the uniform continuity of $f$ that implies the existence of the limit of the 'Riemann sums'; more generally, it implies the existence of a limit for the analogous sums $\sum_i f(\xi_i)(g(x_{i+1}) - g(x_i))$ ($x_i \leq \xi_i \leq x_{i+1}$), where $g$ is only assumed to be a bounded function that is increasing on $[a, b]$. This limit, denoted $\int_a^b f(x)\, dg(x)$ and called the Stieltjes integral of $f$ with respect to $g$, may be regarded as the 'weighted sum' of the function $f$ for the measure $\mu$ defined on the set of semi-open intervals $]\alpha, \beta]$ by the formula $\mu([\alpha, \beta]) = g(\beta+) - g(\alpha+)$; it is no longer tied to Differential Calculus as closely as the usual notion of integral (*).

The same is true for the classical 'double' and 'triple' integrals, associated with the measurement of plane areas and volumes, respectively. However, all of these notions of integral are related to each other, not only by their definition, but by the following characteristics: the 'integral' $\mu(f)$ of a continuous numerical function $f$ on a certain compact subset $K$ of the line, the plane, or 3-dimensional space, is a number associated with the element $f$ of the space $\mathscr{C}(K)$ of continuous functions on $K$; $f \mapsto \mu(f)$ is thus a mapping of $\mathscr{C}(K)$ into $\mathbf{R}$ (sometimes called a 'functional') that is: 1° linear (that is, $\mu(\alpha f + \beta g) = \alpha \mu(f) + \beta \mu(g)$ for all scalars $\alpha, \beta$ and continuous functions $f, g$); 2° positive (that is, $\mu(f) \geq 0$ for every continuous function $f \geq 0$).

It is remarkable that, conversely, these two properties suffice to characterize the Stieltjes integrals on an interval $[a, b]$ (F. Riesz's theorem). If this is so it is because, starting with the values of the integral of continuous functions, one can reconstitute the measure that gave birth to it. This amounts (if we think of the interpretation of $\int_a^b f(x)\, dx$ as a plane area) to calculating the integral of a characteristic function of an interval, assuming it to be known for continuous functions. In other terms, it is a question of extending in a suitable way the functional $\mu(f)$ to a set of functions con-

(*) If, in particular, one takes $g$ to be a step function that is increasing and right-continuous, then the corresponding Stieltjes integral is none other than the weighted sum of $f$ for the point masses $m_i = g(a_i+) - g(a_i-)$ placed at the points of discontinuity $a_i$ of $g$.

taining $\mathscr{C}(K)$ and large enough to contain also the characteristic functions of intervals.

There are several methods for accomplishing this extension; one of the most interesting appeals to the notion of function space. One knows that, on the space $\mathbf{R}^n$, the norms $\|x\|_\infty = \sup_{1 \leq i \leq n} |x_i|$ and $\|x\|_1 = \sum_{i=1}^n |x_i|$ define the same topology. By 'passing from finite to infinite' one is led to consider, on the space $\mathscr{C}(K)$ of continuous functions on a compact interval $K = [a, b]$ of $\mathbf{R}$, the norms $\|f\|_\infty = \sup_{x \in K} |f(x)|$ and $\|f\|_1 = \int_a^b |f(x)| dx$ (or $\int_a^b |f| dg$ in the case of a Stieltjes integral). But here, the topologies defined by these two norms are different, and the space $\mathscr{C}(K)$, which is complete for the first norm (GT, X, §1, No. 6, Cor. 1 of Th. 2), is no longer so for the second. More precisely, one can identify the elements of the completion of $\mathscr{C}(K)$ for the norm $\|f\|_1$ with classes of not necessarily continuous functions, and the extension of the integral is then made simply by extending by continuity the functional $\mu(f)$ defined on $\mathscr{C}(K)$, to the completion of this space (the technical details of this procedure are exposed in Ch. IV). Of course, we have assumed that the integral of continuous functions was defined starting with a measure (by the procedure of 'Riemann sums' reviewed above); to obtain F. Riesz's theorem, it is necessary to operate in the same way, but by defining the norm to be $\mu(|f|)$, where $\mu(f)$ is the positive linear functional defined on $\mathscr{C}(K)$.

The method of extension we have just sketched not only leads to Riesz's theorem, but in addition it permits defining the integral for classes of functions 'much more discontinuous' than the characteristic functions of intervals; on considering the characteristic functions of sets that are 'integrable' functions, it permits, at the same stroke, extending to the corresponding sets the measure given initially only for intervals, by setting $\mu(A) = \mu(\varphi_A)$; this extension of course preserves the fundamental properties of additivity and positivity of the measure.

The foregoing deals with the Stieltjes integral on the line, but the method of extension carries over at once to measures defined in the plane or in space, or on curves or surfaces. More generally, on analyzing the proofs, one perceives that they are actually valid for every positive linear functional defined on the space $\mathscr{K}(X)$ of continuous functions on an arbitrary locally compact space $X$, each of which is zero outside a compact subset (depending on the particular function).

This category of spaces to which the theory of integration is therefore applicable includes naturally the numerical spaces $\mathbf{R}^n$ as well as manifolds; it also includes the discrete spaces (where the theory of integration merges with that of summable families of real numbers (GT, IV, §7)), as well as the products (finite or infinite) of compact spaces identical to an interval of $\mathbf{R}$ or to a finite set; we shall see later on that the theory of measure on such products plays an important role in the Calculus of Probability.

The extension of the concept of measure to general locally compact spaces has shown itself to be especially fertile in the theory of locally compact groups; generally speaking, the notion of integral appears to be the right tool whenever, in Topological Algebra, one wants to 'pass from the finite to the infinite', that is, to generalize the procedures of pure algebra in which finite sums appear, to the case that the 'summation' must deal with an infinite number of terms. For example, one knows (A, III, §2) that the elements of the algebra of a finite group $G$ (over the field $\mathbf{R}$) are the mappings $s \mapsto \alpha(s)$ of $G$ into $\mathbf{R}$, with the multiplication law $\alpha * \beta = \gamma$, where $\gamma$ is the function defined by

What appears to be a natural generalization of this algebra, for an arbitrary locally compact group $G$, is the set of mappings of $G$ into $\mathbf{R}$ that are integrable for a certain special measure $\mu$ on $G$ (the 'Haar measure'), the multiplication in the algebra being given by

Moreover, once embarked on this path, one is quickly annoyed by the obligation to 'sum' only functions with real values; in many cases, it is useful to know how to define the integral of functions that are defined on $X$ and take values in a topological vector space over $\mathbf{R}$, for example a Banach space or a space of operators on a Banach space. One ascertains that this extension can be made easily with no need for any profound modifications of the theory of integration.

In the foregoing sketch, a preponderant role has been given to continuous functions; it is natural to ask whether the notion of measure is in effect tied in an essential way to the existence of a topology on the set $X$ where it is defined. A close examination of the theory shows that this is not at all the case, and that the methods of extension apply as well to a positive linear functional $\mu(f)$ defined on a vector space $\mathscr{V}$ consisting of numerical functions defined on an arbitrary set $X$, by means of certain supplementary conditions imposed on $\mathscr{V}$ and on $\mu(f)$; these conditions are automatically satisfied when $\mathscr{V}$ is a space $\mathscr{K}(X)$ of continuous functions with compact support, but they are also satisfied in more general cases. However, this greater generality is in some respects illusory: indeed, it can be shown that every 'abstract measure' is, in a certain sense, 'isomorphic' to a measure defined (starting with continuous functions) on a suitable locally compact space; on the other hand, most applications have to do with sets X equipped with a topology that intervenes naturally in the matter; we shall therefore occupy ourselves exclusively, until Chapter IX, with measures defined on locally compact spaces.

The first two chapters are preliminaries to the theory: they are devoted to the proof of inequalities fundamental to the extension, and to the study of certain ordered vector spaces, the Riesz spaces, which play an important role in several questions later on.

The concept of measure on a locally compact space is defined in Chapter III; we take as point of departure the theorem of Riesz, which thus becomes a definition: the integral of continuous functions is therefore defined before the measure of sets, as a positive linear functional on $\mathscr{K}(X)$. This presentation offers certain technical advantages (due notably to the fact that the continuous functions form a vector space, whereas this is not the case for the characteristic functions of sets); moreover, it is in the form of a functional on $\mathscr{K}(X)$ that the integral naturally arises in numerous questions. Finally, the differences of two positive linear functionals on $\mathscr{K}(X)$ (which we again call measures on X) may be characterized as the linear forms on $\mathscr{K}(X)$ satisfying certain continuity conditions; the theory of integration is thus related, on the one hand to the general theory of duality in topological vector spaces (cf. Book V) and on the other hand to the theory of distributions, which generalize certain aspects of the concept of measure and which we shall expose in a later Book.

Chapter IV is devoted to the extension of the integral; both integrable functions and the measure of sets are defined there, as well as the function spaces $L^p$, whose importance in applications is considerable; also shown there, is how the introduction of the concept of measurable function leads to convenient criteria for integrability.

In the next two chapters, it will be seen how measurable functions appear also as 'densities', enabling one to define new measures on a space X starting from a given measure. This study, which leads in particular to important results in the duality theory of the $L^p$ spaces, is also tied to the notion of vectorial measure, which can, in the most favorable cases, be brought under the theory of integration (with respect to a positive measure) of vector-valued functions.

We shall also develop what may be considered the modern culmination of the idea of 'decomposition into slices' of plane areas and of volumes, introduced by the founders of the Integral Calculus: under certain conditions, a measure on a space X can be decomposed into a 'sum' of measures each of which is carried by a 'slice' of the space X (that is, by an equivalence class with respect to a certain relation R); moreover, such a decomposition permits calculating the integral of a function, with respect to the original measure, by first integrating 'over each slice', then integrating (with respect to a suitable measure) the resulting function on the quotient space X/R (a generalization of 'double summation' for the sum of a family where the indices run over a product set).

Chapter VII is devoted to the study of Haar measure on a locally compact group, which is characterized, up to a scalar multiple, by the property of being invariant under every left translation of the group.

In Chapter VIII the convolution of measures is exposed, a concept that plays a role of the first order in modern Functional Analysis.

Chapter IX is devoted to integration in Hausdorff topological spaces that are not necessarily locally compact, and in particular in locally convex vector spaces; this permits, notably, the extension of the theory of the Fourier transformation to the latter spaces. The mode of exposition chosen in the early sections consists in reducing, to the extent possible, to the case of compact spaces treated in the earlier chapters.