24. Integration in non locally compact spaces.

If the study of the links between topology and measure theory goes back to the beginnings of the modern theory of functions of real variables, it is only very recently that integration in discrete topological spaces has been set up in a general way. Before giving the history of the work which preceded the actual synthesis, we will recollect some of the steps in the evolution of the ideas concerning the relations between topology and measure.

For Lebesgue, it is only a question of integrating the functions of one or several real variables. In 1913, Radon defined general measures on $\mathbf{R}^n$ and the corresponding integrals; this theory is expounded in detail in the work [82] of Ch. de la Vallée Poussin and relies in a constant way on the topological properties of Euclidean spaces. A bit later, in 1915, Fréchet defines in [115 c] "abstract" measures on a set furnished with a $\sigma$-algebra and the integrals with respect to these measures; he notes that it is possible to establish thus the main results of Lebesgue's theory without using any topological means. He justifies his undertaking with the following words, taken from the introduction to [115 d], first part: "That for example in a space with an infinity of co-ordinates where various applications of Analysis had led to various non equivalent definitions of a convergent sequence, one of these definitions is replaced by another, nothing will be changed in the properties of the families and additive functions of sets in these spaces". The research of Fréchet is completed by Carathéodory, to whom is due the important theorem on the extension of a function of sets into a measure. The beginning of the book of Saks [268] offers a condensed exposition of this point of view.

The discovery of Haar measure on locally compact groups (pp. 231 ff.) and the numerous applications that it receives immediately, then the work of Weil and Gelfand in Harmonic Analysis, lead in 1940 to a deep modification of this point of view: in this kind of question, it is most convenient to consider a measure to be a linear form on a space of continuous functions. This method forces a restriction to compact or locally compact spaces, but it is not a hindrance for almost the totality of the applications; even better, the introduction of Harmonic Analysis on the $p$-adic groups and the adèle groups by J. Tate and A. Weil allowed a spectacular renewing of analytic Number theory.

It is from quite another direction that the necessity comes to enlarge this point of view by the consideration of measures on topological spaces which are not locally compact: little by little, Probability Theory leads to the study of such spaces and supplies numerous non trivial examples. Perhaps the reason for the late influence of these developments on measure theory must be sought in the relative isolation of Probability Theory, which remained on the margin of traditional mathematical disciplines until recent times.

MEASURES ON SEQUENCE SPACES.

One of the most developed branches of classical Probability Theory is that of limit theorems (the law of large numbers, tendency towards the Gauss-Laplace law, ...); it is a case of a deepening of the notion of statistical regularity manifested by phenomena bringing into play very large populations. The correct mathematical formulation of these problems requires the introduction of measures on the space of sequences; these spaces, which constitute the most obvious generalisation of spaces of finite dimension, are the favourite subject of the researches on "General Analysis" undertaken around 1920 by Fréchet, Lévy, Lusin, .... It is in any case not fortuitous that Khintchine and Kolmogoroff, the creators of the new methods in Probability Theory, are both disciples of Lusin, and that Lévy very soon turned to probabilistic problems: these constituted the touch stone of the new methods.

The first implicit occurrence of a measure on a sequence space appears in the work devoted by E. Borel in 1909 to countable probabilities [32 b]. A very original idea of Borel consisted of the application of probabilistic results that he had just obtained to the proof of properties possessed by the decimal expansion of nearly every real number contained between 0 and 1. This application rests on the following fundamental remark: let us define every real number contained between 0 and 1 by the sequence of digits in its expansion in a given base $q$ ($q \geq 2$); if one takes at random successively the different digits from a number $x$, independently the ones from the others and with an equal probability $1/q$ for $0, 1, \ldots, q-1$, the probability that $x$ is to be found in an interval of $[0, 1[$ is equal to the length of this interval.

In 1923, Steinhaus [293] establishes these results rigorously and describes the precise mathematical model of the unbounded sequence of selections of the kind considered by Borel: let us take $q = 2$ to simplify matters and denote by $I$ the set with two elements $\{0, 1\}$; we furnish $I$ with the measure $\mu$ defined by $\mu(0) = \mu(1) = \frac{1}{2}$; the elements of the product space $I^{\mathbf{N}}$ are the sequences $\varepsilon = (\varepsilon(n))_{n \in \mathbf{N}}$ of numbers equal to 0 or 1 and the map $\phi : \varepsilon \mapsto \sum_{n \geq 0} \varepsilon(n).2^{-n-1}$ is, up to a countable set, a bijection of $I^{\mathbf{N}}$ on the interval $[0, 1]$; what is more, $\phi^{-1}$ transforms the Lebesgue measure on $[0, 1]$ into the measure $P$ on $I^{\mathbf{N}}$ the product of the measures $\mu$ on each of the factors. In fact Steinhaus does not have at his disposal a construction for product measures;

he uses the existence of the quasi-bijection $\phi$ to construct the measure $P$ on $I^{\mathbf{N}}$ starting from the Lebesgue measure on $[0, 1]$, then he gives an axiomatic characterisation of $P$. The isomorphism thus obtained allows the translation of the language of probability into that of measure and to apply the known theorems about the Lebesgue integral.

In the same work, Steinhaus considers the random series $\sum_{n \geq 0} \sigma_n \cdot a_n$, where the signs $\sigma_n = \pm 1$ are chosen at random independently the ones from the others and with the same probability $\frac{1}{2}$; between 1928 and 1935, he studies numerous other random series. On their side, Paley, Wiener and Zygmund consider random Fourier series$^1$ of the form $\sum_{-\infty}^{\infty} a_n \exp(2\pi i(nt + \Phi_n))$; the "amplitudes" $a_n$ are fixed, and the "phases" $\Phi_n$ are independent random variables uniformly spread over $[0, 1]$. If the analytic difficulties vary enormously from problem to problem, the translation in terms of measure theory is the same in all cases and represents an extension of the case treated by Borel and Steinhaus; it is a matter of constructing a measure on $\mathbf{R}^{\mathbf{N}}$, the product of a family of measures all identical to one particular positive measure $\mu$ of mass 1 on $\mathbf{R}$; for example, the preceding random Fourier series correspond to the case where $\mu$ is the Lebesgue measure on $[0, 1]$.

To construct such product measures, two methods can be used. The first is a direct method, set up for the first time by Daniell [76 b] in 1918; it is found again in 1934 by Jessen [173] who will make a detailed study of the case where $\mu$ is the Lebesgue measure on $[0, 1]$.. The second method is the search for tricks analogous to those of Steinhaus in order to reduce to the case of the Lebesgue measure on $[0, 1]$; this way of proceeding had the advantage of convenience as long as there is not at our disposal a complete account of the general theory of measure, for it allows the use without new proofs of the Lebesgue theorems.$^2$

THE THEORY OF BROWNIAN MOVEMENT.

This theory occupies an exceptional place in contemporary scientific development because of the constant and fruitful exchange which it witnesses between physical problems and "pure" mathematics. The study of brownian movement, discovered in 1829 by the botanist Brown, has been carried on intensively in the XIXth century by numerous physicists,$^3$ but the first satisfactory mathematical model was invented by Einstein only in 1905. In the simple case of a particle moving along a straight line, the fundamental hy-

$^1$ For an account of random Fourier series, see [176].

$^2$ Wiener also takes care on many occasions (cf. for example [243], chap. IX) to show that the measure of the brownian movement is isomorphic to the Lebesgue measure on $[0, 1]$. The possibility of such tricks finds its explanation in a general theorem of von Neumann who gives an axiomatic characterisation of measures isomorphic to the Lebesgue measure on $[0, 1]$.

$^3$ A very lively account of this history will be found in [229].

potheses of Einstein are formulated thus: if $x(t)$ is the abscissa of the particle at time $t$, and if $t_0 < t_1 < \ldots < t_{n-1} < t_n$, the successive displacements $x(t_i) - x(t_{i-1})$ (for $1 \leq i \leq n$) are independent gaussian random variables. It is not the place to evoke here in detail the important experimental work of J. Perrin which motivated Einstein’s theory: for our purposes, it is sufficient to remember only a remark of Perrin, according to which the observation of the trajectories of brownian movement suggested irresistibly to him “the functions without derivative of the mathematicians”. This remark was the initial spark for Wiener.

Quite another stream of ideas draws its origins from the kinetic theory of gases, developed between 1870 and 1900 by Boltzmann and Gibbs. Let us consider a gas made up of $N$ molecules of mass $m$ at (absolute) temperature $T$ and let us denote by $\mathbf{v}_1, \ldots, \mathbf{v}_N$ the velocities of the $N$ molecules of the gas; the kinetic energy of the system is equal to

where $k$ is Boltzmann’s constant. According to the ideas of Gibbs, the multitude of collisions between molecules does not allow the precise determination of the velocities of the molecules, and it is convenient to introduce a probability law $P$ on the sphere $S$ of the space with $3N$ dimensions defined by the equation (24.1). The “microcanonical” hypothesis consists in assuming that $P$ is the measure of mass 1 invariant under rotation on the sphere $S$. Besides, Maxwell’s law of velocities states that the law of probability of the component of the velocity of a molecule has a Gaussian measure of variance $2kT/m$. E. Borel seems to have been the first to remark in 1914 that Maxwell’s law is a consequence of the hypotheses of Gibbs and of properties of the sphere when the number of molecules is very large. He considers a sphere $S$ in Euclidean space of large dimension and the measure $P$ of mass 1 invariant under rotation on $S$; using the classical methods of approximation based on the Stirling formula, he shows that the projection of $P$ on a co-ordinate axis is approximately Gaussian. These results are made precise a little later by Gâteaux and Lévy [201 a]. Given an integer $m \geq 1$ and a number $r > 0$, let us denote by $S_{m,r}$ the set of sequences of the form $(x_1, \ldots, x_m, 0, 0, \ldots)$ with $x_1^2 + \ldots + x_m^2 = r^2$; let us denote also by $\sigma_{m,r}$ the measure of mass 1 invariant under rotation on $S_{m,r}$. Stated in modern language, the result of Gâteaux and Lévy is the following: the sequence of measures $\sigma_{m,1}$ tends strictly to the unit mass at the origin $(0, 0, \ldots)$ and the sequence of measures $\sigma_{m, \sqrt{m}}$ tends strictly to a measure $\Gamma$ of the form

($\gamma$ is the Gaussian measure with variance 1 on $\mathbf{R}$).

The preceding measure $\Gamma$ plays the role of a Gaussian measure in infinite dimensions. It appears strongly that Lévy had confusedly hoped to define in an intrinsic manner a Gaussian measure on all Hilbert spaces of infinite dimension. In fact, as was shown by Lévy and Wiener, the measure $\Gamma$ is invariant in a certain sense$^4$ under the automorphisms of $l^2$; unfortunately, the set $l^2$ of square summable sequences $(x_1, x_2, \ldots, x_n, \ldots)$ is of measure zero for $\Gamma$. It is known today that we must be content with a Gaussian premeasure on a Hilbert space of infinite dimension.$^5$

To Wiener is owed the essential progress: if there is no reasonable Gaussian measure on a Hilbert space of infinite dimension, it is possible to construct by means of the primitive operation a measure $w$ on a space of continuous functions starting with a Gaussian premeasure. We shall explain succinctly the initial construction of $w$ by Wiener [336]; it is directly influenced by the relation $\Gamma = \lim_{m \to \infty} \sigma_{m, \sqrt{m}}$ of Gâteaux and Lévy. For every integer $m \geq 1$, let us denote by $H_m$ the set of functions on $T = ]0, 1]$ which are constant on each of the intervals $]\frac{k-1}{m}, \frac{k}{m}]$ (for $k = 1, 2, \ldots, m$), and $\pi_m$ the measure of mass 1 invariant under rotation on the Euclidean sphere of radius 1 in $\mathbf{R}^m$. Let $f_m$ be the isomorphism of $H_m$ on $\mathbf{R}^m$ which associates with each function taking the value $a_k$ on the interval $]\frac{k-1}{m}, \frac{k}{m}]$ the vector $(a_1, a_2 - a_1, \ldots, a_m - a_{m-1})$ (whence the name of "differential space" dear to Wiener); let us denote by $w_m$ the measure on $H_m$ image of $\pi_m$ under $f_m^{-1}$. Wiener defines the measure $w$ being sought as the limit of the measures $w_m$. Precisely, let us denote by $H$ the set of regular functions on $T$, with the uniform convergence topology (we have $H_m \subset H$ for every integer $m \geq 1$); for every uniformly continuous function $F$ bounded on $H$, the limit $A\{F\} = \lim_{m \to \infty} \int_{H_m} F(x) dw_m(x)$ exists; Wiener then obtains certain majorations by a subtle analysis of the fluctuations of the items in play, and, taking up again the arguments of compactness brought out by Daniell, he shows that we are in a situation for the application of Daniell's extension theorem. We conclude the existence of a measure $w$ carried by $C(T)$ and such that $A\{F\} = \int_{C(T)} F(x) dw(x)$. Wiener can then show that the measure $w$ corresponds to the hypotheses of Einstein,$^6$ and his estimates allowed him to give a precise

$^4$ Precisely, the following result holds. Let $U$ be an automorphism of the Hilbert space $l^2$ and $(u_{mn})$ the matrix of $U$. Let $E$ be the vector space of all real sequences $(x_n)_{n \geq 1}$ and $F$ the subspace of $E$ made up of the sequences $(x_n)_{n \geq 1}$ for which the series $\sum_{n \geq 1} u_{mn} x_n$ converges for all $m \geq 1$. The formula $(Ux)_m = \sum_{n \geq 1} u_{mn} x_n$ defines a linear map $\tilde{U}$ of $F$ into $E$, the measure $\Gamma$ is concentrated on $F$ and we have $\tilde{U}(\Gamma) = \Gamma$.

$^5$ This notion was introduced under the name of "weak canonical distribution" by I. Segal [281]. To this author is owed a detailed study of the Gaussian premeasures, and their application to certain problems of the quantum theory of fields.

$^6$ This is translated by the formula

meaning to the remark of Perrin on functions without a derivative: the set of functions satisfying a Lipschitz condition of order $\frac{1}{2}$, is negligible for $w$ (on the contrary, for every $a$ with $0 < a < \frac{1}{2}$, nearly every function satisfies a Lipschitz condition of order $a$).

We know today numerous constructions for the Wiener measure. Thus, Paley and Wiener use random Fourier series ([243], chap. IX): to every real sequence $a = (a_n)_{n \geq 1}$ and every integer $m \geq 0$, let us define the function $f_{m,a}$ on $]0,1]$ by

it can be shown that, for $\Gamma$-almost all sequences $a$, the sequence of functions $f_{m,a}$ tends to a continuous function $f_a$ and that $w$ is the image of $\Gamma$ by the map (defined almost everywhere) $a \mapsto f_a$. Later, Lévy gave in [201 b and c] another construction. Finally Kac, Donsker and Erdös show around 1950 how to replace in Wiener’s initial construction the spherical measures $\pi_m$ on $\mathbf{R}^m$ by more general measures. Their results establish a solid link between Wiener’s measure and the limit theorems of Probability Theory; they will be completed and systematised by Prokhorov [254] in work to which we will return later.

This is not the place to analyse the numerous and important probabilistic works caused by Wiener’s discovery; today, the brownian movement appears only as one of the most important examples of a Markov process. We will only mention the application made by Kac of Wiener’s measure to the solution of certain parabolic partial differential equations; it is a matter there of an adaptation of the ideas of Feynman in quantum field theory — another example of this reciprocal influence between mathematics and the problems of physics.

PROJECTIVE LIMITS OF MEASURES.

This is the case of a theory that was developed especially in view of the needs of Probability Theory. The problems concerning a finite sequence of random variables $X_1, \ldots, X_n$ are solved in principle when the law $P_X$ of this sequence is known: it is a positive measure of mass 1 on $\mathbf{R}^n$, such that the probability of obtaining simultaneously the inequalities $a_1 \leq X_1 \leq b_1, \ldots, a_n \leq X_n \leq b_n$ is equal to $P_X(C)$ where $C$ is the closed box $[a_1, b_1] \times \ldots \times [a_n, b_n]$ of $\mathbf{R}^n$.

(we have a convention that $x^0 = 0$). Wiener, trained in analytic rigour by Hardy, and justifiably wary with regard to the foundations of Probability Theory at that time, takes care not to use either the terminology or the probabilistic results. As a result his memoirs are full of formidable formulae of which the preceding one is a sample; this peculiarity is one of the factors that slowed down the diffusion of Wiener’s ideas.

In practice, the measure $P_X$ has discrete support or else admits a density with respect to Lebesgue measure. When it is the case of an infinite sequence $(X_n)_{n \geq 1}$ of random variables, the law $P_n$ of the partial sequence $(X_1, \ldots, X_n)$ is known in general for every integer $n \geq 1$; these given laws satisfy a compatibility condition that expresses the fact that the sequence $(P_n)_{n \geq 1}$ is a projective system of measures. Until around 1920, one defined in a more or less implicit manner the probabilities of events linked to the infinite sequence by a “natural” passing to the limit starting with probabilities in the finite case; thus it will be assumed that the probability that a game stops is the limit, for $n$ tending to infinity, of the probability that it stops in at most $n$ steps. Naturally, such a theory is hardly coherent, and nothing excludes the presence of “paradoxes”, a same probability receiving two distinct estimations according to whether it is evaluated by one or other of two procedures each as “natural” as the other.

Steinhaus [293] seems to have been the first to feel the necessity of considering (for the case of the toss of a coin) not only the projective system $(P_n)_{n \geq 1}$ but its limit. Shortly before, in 1919, Daniell [76 d] had proved the existence of such projective limits in general,$^7$ but this result seems to have remained unknown in Europe. It is found again in 1933 by Kolmogoroff in the work [183] where this author formulates the axiomatic conception of Probability Theory. The proofs of Daniell and Kolmogoroff use a compactness argument that rests on Dini’s theorem.

The Daniell-Kolmogoroff theorem leaves nothing to be desired in the case of random sequences $(X_n)_{n \geq 1}$, but the study of random functions undertaken starting in 1935 by Kolmogoroff, Feller and Doob conceals difficulties of quite another order. Let us consider for example an interval $T$ of $\mathbf{R}$, which represents the set of instants of observations of a “stochastic process”; the set of possible trajectories is the product space $\mathbf{R}^T$, considered as a projective limit of partial products $\mathbf{R}^H$, where $H$ runs through the set of finite subsets of $T$; one obtains in general a projective system of measures $(\mu_H)$. Kolmogoroff’s theorem does indeed provide a measure on $\mathbf{R}^T$, but it is defined on a family that is notably smaller than the Borel family.$^8$ A variant of Kolmogoroff’s construction, which provides a measure on a topological space, is due to Kakutani [177], and has been rediscovered many times since: one considers $\mu_H$ as a measure on $\overline{\mathbf{R}}^H$ carried by $\mathbf{R}^H$;$^9$ the compact space $E = \overline{\mathbf{R}}^T$ is a projective limit of the finite products $\overline{\mathbf{R}}^H$ and a measure $\mu$ on $E$ can be

$^7$ Daniell deals with the case of measures on a product $\prod_{n \geq 1} I_n$ of compact intervals of $\mathbf{R}$, but his method extends immediately to the case of an arbitrary product of compact spaces.

$^8$ Kolmogoroff’s measure is only defined for Borel sets in $\mathbf{R}^T$ of the form $A \times \mathbf{R}^{T-D}$ where $D$ is a countable subset of $T$, and $A$ is a Borel subset of $\mathbf{R}^D$; from this, Kolmogoroff’s theorem for an arbitrary product $\mathbf{R}^T$ is an immediate consequence of the case for countable products.

$^9$ It would be possible to replace $\overline{\mathbf{R}}$ by any compact space containing $\mathbf{R}$ as a dense subspace.

defined as a projective limit of the $\mu_H$. But this procedure has a severe inconvenience; the elements of $\overline{\mathbf{R}}^T$ do not possess any regularity property allowing the continuation of the probabilistic study of the process — or even simply the elimination of the parasitic values $\pm \infty$ introduced by the compactification $\overline{\mathbf{R}}$ of $\mathbf{R}$. It can be remedied by restricting the measure $\mu$ of $\overline{\mathbf{R}}^T$ to a particular subspace (for example $\mathscr{C}(T)$ in the case of brownian movement); the fundamental difficulty arises from the fact that a functional space, even of the usual type, is not necessarily $\mu$-measurable in $\overline{\mathbf{R}}^T$, and even the choice of the functional space can be in question.$^{10}$

A decisive step was accomplished in 1956 by Prokhorov in a work [254] which exercised a determining influence on the theory of stochastic processes. By putting into a convenient axiomatic form the methods used by Wiener in the article analysed above, he establishes a general existence theorem for projective limits of measures on functional spaces.

A more restricted class of projective systems was introduced by Bochner [24 b] in 1947; it is a case of projective systems made up of real vector spaces of finite dimension and surjective linear transformations. The projective limit of such a system is identified in a natural way with the algebraic dual $E^*$ of a real vector space $E$ supplied with a weak topology $\sigma(E^*, E)$; a corresponding projective system of measures has a limit which is a measure $\mu$ defined for a family which is notably smaller than the Borel family of $E^*$. Bochner characterised completely such "premeasures" by their Fourier transform, which is a function on $E$. But this result is hardly usable in the absence of a topology on $E$, in which case the possibility of considering $\mu$ as a measure on the topological dual $E'$ of $E$ must be examined. In an independent way, R. Fortet and E. Mourier, while seeking to generalise to random variables with values in a Banach space certain classical results of Probability Theory (the law of large numbers, the central limit theorem) brought out also the fundamental role played by the Fourier transform in these questions. But substantial progress was only made in 1956 when Gelfand [125 c] suggested that the natural framework for the Fourier transform is not that of Banach spaces or Hilbert spaces, but that of nuclear Fréchet spaces. He conjectured that every continuous function of positive type on such a space is the Fourier transform of a measure on its dual, a result established soon after by Minlos [222]. Its importance comes above all from the fact that it is applicable to distribution spaces, and that the quasi-totality of functional spaces are Borel subspaces of the space of distributions (which thus constitutes a much better setting than $\mathbf{R}^T$).$^{11}$ The theory of random distributions is an area in full expansion, and we will be content with referring the reader to the work of Gelfand and Vilenkin [126].

$^{10}$ For a detailed discussion of the problem of construction of measures for functional spaces, and the methods used before Prokhorov, see J. L. Doob [93].

$^{11}$ The exposition by X. Fernique [110] can be consulted, which also contains numerous results on strict convergence.

24. INTEGRATION IN NON LOCALLY COMPACT SPACES.

The results that we have just mentioned on projective limits use the existence of topologies on the base spaces. It can be asked if there exists an analogous theory in the case of "abstract" measures. Von Neumann proved already in 1935 the existence of product measures in all cases, but the discovery of a counterexample by Jessen and Andersen [290] ruined the hope that every projective system of measures admits a limit. Two palliatives were discovered: in 1949, C. Ionescu-Tulcea established the existence of countable projective limits, granted the existence of suitable decompositions,$^{12}$ a very interesting result for the study of Markov processes; besides, it became clear that the topology of the spaces does not come in except by means of the set of compact subsets. It was therefore natural to seek to axiomatise this situation inside an abstract theory, by means of the notion of a compact class of subsets of a set. This work was done in 1953 by Marczewski (who established by this means a theorem on abstract projective limits) and Ryll-Nardzewski (who worked on the decomposition of measures).$^{13}$

MEASURES ON GENERAL TOPOLOGICAL SPACES AND STRICT CONVERGENCE.

The study of the links between topology and measure theory was especially thought of as the study of regularity properties of measures, and in particular that of "exterior" regularity and of "interior" regularity;$^{14}$ interior regularity is equivalent to exterior regularity on a locally compact space countable at infinity. The construction that Lebesgue gives of the measure of sets on the straight line brings out these two kinds of regularity, and the property of exterior regularity of measures on a Polish space seems to have had public notoriety around 1935. But it is only in 1940, in an article whose diffusion was slowed down by the war, that A. D. Alexandroff [3] brought out the role of interior regularity and showed that this is possessed by the measures on a Polish space; this result was found again later by Prokhorov [254] and is often attributed mistakenly to this author. It only appeared very recently that this property extended to sublinian spaces; because of this, the importance of these spaces increased greatly, all the more so that it became known that their

$^{12}$ It appears that it is the absence of a suitable theory of decompositions that marks the limit of the theory of "abstract" measures. This difficulty reappears in an insistent way in Probability Theory in connection with conditional probabilities. $^{13}$ For an account of this theory, one can refer to [249]. $^{14}$ An "abstract" measure $\mu$ on the Borel family of a discrete topological space is said to be exterior regular if the measure of every Borel set is the lower bound of the measures of the open sets that contain it; the measure $\mu$ is said to be interior regular if the measure of every Borel set is the upper bound of the measure of its compact subsets.

theory could be done without a metrisability hypothesis, and that the quasi-totality of functional spaces were sublinian (and even most often lusinian).15

The definition of a mode of convergence (weak or strict) for measures is done in the most convenient way by putting in duality the space of measures with a space of continuous functions. Generalising an ancient result of F. Riesz, A. A. Markhoff established in 1938 a bijective correspondence between the positive functionals on $C(X)$ and the regular measures on a compact space $X$. In the work [3] already quoted, A. D.. Alexandroff extends these results to the case of a completely regular space: he introduces a hierarchy on the set of positive linear forms on the space $C^b(X)$ of bounded continuous functions on a completely regular space $X$,16 he defines strict convergence of bounded measures and proves among others the following two theorems:

a) if $X$ is Polish, the set of linear forms on $C^b(X)$ corresponding to measures is closed in the weak convergence of sequences;

b) if a sequence of bounded measures has a strict limit, "there is no mass fleeing to infinity" (it is a weak form of the reciprocal of the theorem of strict convergence of Prokhorov).

From this fusion of notions and theorems, Prokhorov will know how to extract important results for the theory of stochastic processes, and to present them in a simple and striking form. In his great work of 1956 already quoted [254], an important part is devoted to bounded positive measures on a Polish space; in generalising a construction of Lévy, he defines on the set of positive measures of mass 1 a distance which makes it into a Polish space, then he establishes an important compactness criterion for strict convergence. Independently of Prokhorov, Le Cam [197] has obtained a certain number of compactness results for the strict convergence of measures; he puts no metrisability hypotheses on the spaces that he considers, and his results reduce to previous theorems of Dieudonné in the locally compact case.

15 In order to solve certain probabilistic difficulties (particularly the links between different notions of stochastic independence or dependence) several authors introduce restricted classes of "abstract" measures: the "perfect" spaces of Kolmogoroff-Gnedenko, the "lusinian" spaces of Blackwell, the "Lebesgue" spaces of Rokhlin. In fact (at least granted a fairly weak countability hypothesis), all these definitions give characterisations of "abstract" measures isomorphic to a positive bounded measure on a sublinian space. [249] can be consulted on this subject.

16 He distinguishes by means of a decreasing order of generality the $\sigma$-measures ("abstract" measures on the Borel family of $X$), the $\tau$-measures (exterior regular measures) and the stiff measures (interior regular measures). When $X$ is Polish, these three notions coincide. The terminology itself is due to MacShane and Le Cam [197]. An account of the work arising from this classification will be found in [316].