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Theory of Capacities Annales De L’Institut Fourier, Tome 5 (1954), P ANNALES DE L’INSTITUT FOURIER GUSTAVE CHOQUET Theory of capacities Annales de l’institut Fourier, tome 5 (1954), p. 131-295 <http://www.numdam.org/item?id=AIF_1954__5__131_0> © Annales de l’institut Fourier, 1954, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THEORY OF CAPACITIES (l) by Gustave CHOQUETOQ. INTRODUCTION This work originated from the following problem, whose significance had been emphasized by M. Brelot and H. Cartan : Is the interior Newtonian capacity of an arbitrary borelian subset X of the space R3 equal to the exterior Newtonian capacity of X ? For the solution of this problem, I first systematically studied the non-additive set-functions, and tried to extract from their totality certain particularly interesting classes, with a view to establishing for these a theory analogous to the classical theory of measurability. I succeeded later in showing that the classical Newtonian capacity f belongs to one of these classes, more precisely: if A and B are arbitrary compact subsets of R3, then AAU^+AAflB^/^+AB). It followed from this that every borelian, a^rid even every analytic set is capacitable with respect to the Newtonian capa- city, a result which can, moreover, be extended to the capa- (') This research was supported by the United States Air Force, throught the Office of Scientific Research of the Air Research and Development Command. Ce travail a ete mis au point en anglais a la suite de conferences en anglais; une version francaise n'aurait pu etre prete a temps pour 1'insertion dans ce volume. Le petit article qui suit, completant celui-ci, est naturellement public dans la meme langue. (Note de la Redaction). (2) Visiting Research Professor of Mathematics, University of Kansas, Lawrence, Kansas, 1953-1954. (3) I wish to express my thanks to Professor G. B. Price for the very valuable help which he extended to me in connection with establishing the final version of the English text; thanks are due also to Gr. Ladner, K. Lucas, and E. McLachlan for their untiring collaboration. 132 GUSTAVE CHOQUET cities associated with the Green's function and to other clas- sical capacities. The above inequality, which may be called the inequality of strong sub-additivity, is equivalent to the following: V.(X; A, B)=f{X)—f(X[jA)—f{X[jB)+f{X[j^B)^0. Now, this relation is the first of an infinite sequence of inde- pendent inequalities, each of the form Vn(X; Ai,A2,...,AJ^O, which expresses the fact that the successive differences — in an obvious sense — of the function f are alternately positive or negative. Thus, the Newtonian capacity is seen to be an analogue of the functions of a real variable whose successive derivatives are alternately positive and negative. It is known from a theorem of S. Bernstein that these functions, termed completely monotone, have an integral representation in terms of functions e~^. Likewise, the set functions which are « alternating of order infinity » possess an integral representation in terms of exponentials, that is, of set functions ^(X) which satisfy 0^;^1 and ^(XUY)==^(X).4(Y). These exponentials take values in [0,1] only, and this makes it possible to give a remarkable probabilistic interpretation of the functions which are alternating of order infinity. More generally, a detailed study of several other classes of functions justifies the interest in the determination of the extremal elements of convex cones of functions, and in the utilization of the corresponding integral representations. CHAPTER I. — Borelian and analytic sets in topological spaces. — In this chapter, borelian and analytic subsets of arbitrary Hausdorff spaces are redefined and studied. In fact, a mere adaptation of the classical definitions would lead to sets of an irregular topological character for which an interesting theory of capacitability could not be construc- ted. Therefore, we designate as borelian and analytic sets the sets which are generated by beginning with the compact sets and using the operations of countable intersection and union, and continuous mapping (or projection) only. Thus THEORY OF CAPACITIES 133 the operations of « difference » or « complementation » are not used. The role which the G§ sets play in the classical theory is here played by the K^ sets. CHAPTER II. — Newtonian and greenian capacities. — In this chapter, the Newtonian and Greenian capacities of compact sets and, thereafter, the interior and exterior capa- cities of arbitrary sets, are defined. An equilibrium poten- tial A(X), and a capacity, /*(X), are associated with each compact subset X of a domain. The successive differences (—l^V^X; Ai, ...,AJ are defined for each of these func- tions; it is shown that each Vn is non-positive, and the condi- tions for the vanishing of the Vn are determined. It is shown that the sequence of these inequalities for the capacity f is complete in the sense that every inequality between the capacities of a family of compact sets obtained from p arbitrary compact sets by the operation of union is a consequence of the inequalities Vn ^ 0. A more penetra- ting analysis shows that this result remains valid for the capa- cities of the sets which are obtained from p arbitrary compact sets by means of the operations of union, intersection, and difference. From the relation V2^:0, the following important inequality is obtained : f( U A') -f( W ^ S^A-) -^a-')]' for every finite or countable family of couples of compact sets Of and Ai satisfying the relation a,C:Af for each i. Furthermore, from the relation Va^O, we deduce a result concerning the capacity of certain compact sets, relative to domains which are invariant under a one-parameter group of euclidian motions. The chapter ends with the study of the differential of f(K) with respect to suitable increments AK of K, and with the derivation of a formula which shows that the Green's function G(Pi, Pa) of a domain is a limit of the function G(K K,-fW+f(K,}-f(K,[}K,) ^K.,K,)- 2f(K,)./-(K,) 134 GUSTAVE CHOQUJET CHAPTER III. — Alternating and monotone functions. Capa- cities. — This chapter introduces several classes of func- tions and certain basic concepts as follows: alternating (monotone) functions of order n or oo which are mappings from a commutative semi-group1 into an ordered commu- tative group and which satisfy inequalities of the form Vp^O(Vp^O); the concept of conjugate set functions, connected by the relation P ^(X7) + y(X) = 0, where X' == I X is the complement of X the concepts of capacity on a class of subsets of a topological space, of interior capacity (/*J and exterior capacity (/**), of alternating and monotone capacities, of a capacity which is the conjugate of another capacity. CHAPTER IV. — Extension and restriction of a capacity. — The extension jfa of a capacity /\, defined on a class 81 of subsets of a space E, to a class §3 properly containing §1, by means of the equality /*a(X) == /^(X), can often be used as a means for regularizing the class §1 and also as a means of simplifying proofs of capacitability. On the other hand, the operation of restriction will some- times make it possible to replace the space E by a simpler space. Furthermore, the preservation of various classes of capa- cities (alternating or monotone) under these operations of extension and restriction is studied; for instance, let 84 be the class of all compact subsets of a Hausdorff space E $ then, if /*i is alternating of order n, the same is true for every exten- sion jfi of /a. CHAPTER V. — Operations on capacities and examples of capacities. — First, several operations which leave certain classes of capacities invariant are studied: for instance, if a capacity g(Y), alternating of order n, is defined on the class 3?(F) of all compact subsets of a space F, and if Y = y(X) denotes a mapping from ?t(E) into 3t(F) such that y(X, u Xg) ^ y(Xi) u y(Xa) and which satisfies, in addition, a certain requirement of continuity on the right, then the THEORY OF CAPACITIES 135 function /'(X) == g(Y) is also a capacity which is alterna- ting of order n. The projection operation is such an operation and will play an essential role in the study of capacitability. The remainder of the chapter is devoted to the study of the following important examples of functions and capacities which are alternating of order oo : the operation sup on a group lattice; increasing valuation on a lattice (for example, a non-negative Radon measure); functions derived from a probabilistic scheme; exponentials; energy of the restriction of a measure to a compact set; and others. CHAPTER VI. — Capacitability. Fundamental theorems. — First, the alternating capacities are studied: we establish conditions, to be imposed on E, 8, and /*, which will suffice to insure the preservation of capacitability under finite or denumerable union, and which will imply the validity of the relation /-((J A,) = limf(A,) (where A, c A,,,). It then follows from a general theorem that every K<y§ contained in a Hausdorff space E is capacitable with respect to every alternating capacity f defined on .^(E). In order to pass from these K^ sets to arbitrary borelian and analytic sets, we use the fact that every analytic subset of E is the projection on E of a K^ contained in the product space (E X F), where F is an auxiliary compact space; and we asso- ciate with /*the capacity g on ?t(E X F), where g is defined by g(X)=f(pr.X).
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