VOL. 48, 1962 MATHEMATICS: I. S. GAL 775 is satisfied identically. The same is true of all equations of type (10) and the solution is completed.

THE MARTIN BOUNDARY'* BY ISTVAN S. GkL

UNIVERSITY OF MINNESOTA Communicated by Einar Hille, March 26, 1962 Here we shall study the Martin boundary and its known generalizations from the point of view of uniform structures. Our object is to obtain a topologically satisfactory characterization of these boundaries in terms of the associated uniform structures. We can then use the result to define the Martin boundary of arbitrary spaces possessing Green functions with respect to any given differential operator and an associated boundary-value condition. It is assumed that the reader is familiar with the concept of a uniform structure, its usefulness, and a few fundamental notions such as precompactness and inverse image. The necessary information is available in Kelley's General Topology or in the author's Topological Foundations of Analysis and Geometry. The significance of uniform structures in the definition of Martin boundaries was first pointed out by Brelot in reference 2 and the structure %'M to be defined below is mentioned there. However, Brelot made no attempt to prove that %"EM is identi- cal with the unique uniform structure 'CM associated with the Martin compactifica- tion. He proves only that the compactifications with respect to TEM and VM yield homeomorphic spaces, the question whether a uniform isomorphism indeed exists between these compactifications is left open. Our main result can be summarized as follows: The Martin structure "LM is the weakest uniform structure such that the quotient of any two Green functions K( *; pI, P2) = 9( *; pl)/g( ; P2)

is uniformly continuous with respect to 'lM and the unique uniform structure compatible with the topology of the compactified real line [- , + o ]. This statement is both a theorem and a definition: In the classical case a domain D is compactified by the introduction of a set A of ideal points and by a proper topologization of the set OD = D U A. The Hausdorff space OI admits a unique uniform structure whose trace on D we denote by vM. Then the above proposition states that 'EM = %)M. The same is true in any other situation where a Martin- type compactification has been defined. This includes also the discrete case recently studied by Doob in reference 3. If instead of a domain D we consider a differentiable manifold admitting Green functions, then the above statement be- comes a definition. Let X be a nonvoid set, (1Y, 'V) a uniform space, and 0 a family of functions Downloaded by guest on September 25, 2021 776 MATHEMATICS: I. S. GAL PROC. N. A. S.

(p: X -- Y. Then the weakest uniform structure on X which makes the functions base {s-l(V)} (V e V) consisting of the sets so-1(v) = { (xI, X2): (p(X1), p(X2)) e V}. If CU is precompact, so is every p-'(V) and thus 'U is a precompact structure. (See for example pp. 308-309 of ref. 4.) Since Y = [- c, + co] is compact, its unique uniform structure is precompact, and so our remark can be applied to the family + = I K( P; p2)}, and the corresponding uniform structure c11M* Hence, we obtain the following corollary to our definition: The Martin structure cUM is precompact. As soon as a uniform structure 'U is introduced on a set X, we immediately know the existence of a uniform space (X, 'U.) and an injection map f: X X such that f(X) is dense in X and 'U = f-'(qt). The triple (f, X, ca) is called the completion of X with respect to %U. Up to a uniform isomorphism, the completion is uniquely determined by X and 'U. The enveloping space X is compact if and only if 'U is precompact. Therefore, the foregoing corollary can be expressed also as follows: The completion of the underlying space X with respect to its Martin structure 'uM '/ields a compact space X, the Martin compactification of X. The set aX = X - X is the Martin boundary. We recall that in the classical case and also in any of its extensions a sequence m1, M2, ... of points mi e X converges to a boundary point m e aX if and only if (i) (i = 1, 2, ...) has no accumulation point in X and the numerical sequences (K(mi; Pi, P2)) (i = 1, 2, ...) are convergent for every pair (Pi, P2) e X X X - I. This fact is very important in various applications, e.g., in probability theory. Instead of sequences (mj) (i = 1, 2, . . .), we may consider also nets (mi) (i e I) with values m, E X of filters ~F = {F } with sets F C X. This follows from the fact that all known Martin structures are metrizable and so for every net (in2) (i e I) there is a sequence (in,) (i = 1, 2, ... ) such that (mi) (i e I) and (mi) (i = 1, 2, . . .) converge or diverge simultaneously and in the case of convergence they have the same limit. We are going to prove that if sequences are replaced by nets or filters then the proposition holds for any Martin structure 'aM. It is convenient to formulate it in terms of the Cauchy property in- stead of convergence: A net (mi) (i e I) is a Cauchy net with respect to the Martin structure 'aM if and only -if (K(m1; pi, P2)) (i I) is a Cauchy net in [-a, + o ] fo? every pair (pI, P2) e X X X - I. A similar proposition holds for filters ff instead of nets (mi) (i e I). The only difference is that the family IK(F; PI, p2)1 (F e l) is not the whole image filter K(5V; Pi, P2) but only a filter base. Since filters and nets exhibit the same adherence and convergence properties (see for instance refs. 1 and 5), it is sufficient to prove Downloaded by guest on September 25, 2021 VOL. 48, 1962 MATHEMATICS: I. S. GAL 7,7

the proposition for either one of these. In the present problem, it is easier to handle filters than nets. It turns out that the last proposition holds not only for the Martin structure EM but for any uniform structure 'U of the type 'U. = lub PROPOSITION 1. Let 4 = s4I be a family offunctions

PROPOSITION 2. Let (X, Al) and (Y, VU) be uniform spaces where Y is a Hausdorff space and let f map X into Y such that the image of every Cauchy filter in X is a Cauchy filter in Y. Then f is continuous and it can be continuously extended to a continuous functionf mapping the completion X of X into the completion Y of Y. Proof: For each x e X, we denote by 97v the filter formed by the neighborhoods Ng of x in the space X. The sets Nf = N.fnX are not void because X is dense in X and they form a Cauchy filter 9ix in X. By hypothesis, the sets f(Nj) (N, e 9z1) are the elements of a base for a Cauchy filter f(91,), the image of Ng under f in }T. This filter f(91g) is a base for a convergent filter in the complete space (Y, V), and so we can define f(x) = lim f(91). Since Y is a Hausdorff space, the limit consists of a single point and so the definition is meaningful. If x = x e X and for simplicity y = f(x), then by the definition of the limit for every neighborhood NY there is a neighborhood N, such that f(Nx) C NY. Since Y is a Hausdorff space, we see that f(x) fln Nv = I y} and so f(x)y. Therefore the original value of f at x coincides with limf(%Th) = f(x). Hence f is continuous andf: X -> Y is an extension off: X - Y. The continuity of f at x can be proved as follows: Given a neighborhood N9 of y = f() in the space Y, we wish to find a neighborhood N, of x in X such that f(N1) C N1. We may suppose that Ny is a closed set in Y. By the definition of f(x), there is a neighborhood Ng in X such that f(N) C Ng and so by f(X) C Y one has f(Ng) C N9 = Ny n Y. We can assume that lVg is an open set of X. We prove that No satisfies the requirement: In fact, if t eiNg then given any neighbor- hood No of 11 = f(t) by the definition off(i), there is an open neighborhood NT such that f(N-) C N.,. The sets Nx and No] are open and intersect in t. Hence, X being dense in X, the set S = NX n NT is not void and by the construction f(S) C N-, n N.. This shows that NV n N., and a fortiori l, n N., are nonvoid sets. There- fore, every neighborhood N. of v intersects the closed set N. and so E N, We proved that f(t) = E eT for every eN. Hence, the extension f: X Y is continuous. PROPOSITION 3. Let (X, S1) and (Y, V) be uniform Hausdorff spaces anI let one of them be precompact. Suppose f: X Y is a one-to-one map of X onto Y such that f is a Cauchy filter in X if and only if its image under f isa(Cauchy filter in Y. Then f: X -> Y is a uniform isomorphism. Proof: Let us suppose that the uniform structure %l is precompact. Then the completion X is compact and so the continuous map f: X -Y R whose existence is stated in Proposition 2 is uniformly continuous with respect to the extensions qt and V of the given uniform structures Al and 0c. (See for example Theorem 31, p. 198, in Kelley's book.) Thus, the original map f: X -- Y is uniformly continuous with respect to cU and 0. The mapf: X Y being continuous, Y = f(X) is com- pact and so Vc is precompact. Therefore, the same reasoning can be applied to f-1 and we see thatffis also uniformly continuous. Hence, we have bothf-'(I) < it, i.e., V < f(cl) and f(cl) < V, which show that f(%u) = V. COROLLARY. Up to a uniform isomorphism, each precompact uniform structure is uniquely determined by its Cauchy filters. For example, let us now consider the domains D studied by R. S. Martin in reference 7. In Proposition 3, we let X = Y = D and f be the identity map of D onto itself. As %U we choose the abstract Martin structure TM and for V we put the uniform structure VM introduced by Martin via the metric p(M, M') in Defini- Downloaded by guest on September 25, 2021 VOL. 48, 1962 MATHEMATICS: I. S. GAL 779

tion 2 of his paper. We know that cU is precompact and so by Proposition 3 we get cU = cUM = = vM provided the same filters are Cauchy filters with respect to cU and V. By Proposition 1, a is a Cauchy filter relative to 'I if and only if K(5; Pi, P2) is a Cauchy filter in [-ca, + oo I for every (Pi, P2) e X X X - I. The Cauchy filters of (Y, 1C) were determined essentially by Martin himself: First of all co is a com- plete metric structure and so it is sufficient to determine which elementary filters, i.e. sequences, are convergent to some point of Y = O) = D U A. If the sequence (mn) (i = 1, 2, ... ) has an accumulation point in D, then it is convergent in D if and only if it is convergent to some point m of D. Hence, if and only if K(mi; PI, P2) -- K(m; PI, P2) for every pair (pi, P2) e D X D - I. On pp. 147-148 of reference 7, it is proved that a sequence (ma) (i = 1, 2, ...) having no accumulation points in D is convergent in D if and only if (K(m,; Pi, p2)) is convergent for every pair (pi, P2) E D X D -I. (See the remark at the end of this paper.) Therefore, a sequence (mi) (i = 1, 2, ... ) is a Cauchy sequence relative to VM if and only if each (K(mi; Pi, Ps)) is a Cauchy sequence in [-X, + oo]. Thus, by the metrizability of coM, a filter 5F is a Cauchy filter with respect to 'CM if and only if every K(ff; Pi, P2) is a Cauchy filter in [-X, + c ]. We see that clM and 'cM have the same Cauchy filters and so 'LM = CoM. We add a few final remarks which might be useful to researchers interested in this subject: . First of all, it had been pointed out by Heins on p. 569 of reference 6 and by Parreau on p. 149 of reference 8 that the definitions and proofs given by Martin can be extended without modifications to arbitrary hyperbolic Riemann surfaces. Since Cauchy filters are again characterized by the property that their images under the maps m K(m; P1, P2) are Cauchy filters, we see that the uniform struc- ture introduced in this manner coincides with the abstract Martin structure cuM. More generally, the reasonings presented in this paper imply the validity of the following principle: PROPOSITION 4. Suppose (X, Ct) and (Y, VC) are uniform spaces, VC is precompact and q = { (p} is a family offunctions ~a: X -- Y such that 9f is a Cauchy filter in X if and only if op(g) is a Cauchyfilter in Yfor every (p ee. Then,

Cu = lub { m-(V)}

Looking at Martin's brilliant paper for the first time, one is puzzled by his defini- tion of Cauchy sequences. Although the description of these sequences is incor- rect, it is clear what Martin had in mind: He wanted to give a definition of Cauchy sequences and at the same time he also wanted to point out that at least one ideal boundary point is lying above each Euclidean boundary point of the domain D. One final remark: There seems to be considerable confusion among some prob- abiitist on the subject of "Martin topology" or "fine topology." As far as the domain D (i.e., 1 or X) is concerned, there is no such thing. The topology of D is intact and only its "usual" (Euclidean) uniform structure is replaced by the Martin structure CuM. The completion of D with respect to CuM gives a larger enveloping topological space ) awnd the Wpokg of this space O) is called "fine" by Brelot, Doob, Hunt, and thpir students. The trace of the fine topology on D is the "usual' Downloaded by guest on September 25, 2021 780 MATHEMATICS: P. A. GRIFFITHS PROC. N. A. S.

topology. In the case of the Martin exit boundary of the set £Q = {1, 2, 3, . . . }, the usual topology and uniform structure are discrete, i.e., points form open sets and I is a uniformity. * This research was supported under contract with the Office of Ordnance Research at Yale University. The work was completed at the University of Minnesota. 1 Bartle, R. G., "Nets and filters in topology," Am. Math. Monthly, 62, 551-557 (1955). 2 Brelot, M., "Sur le principe des singularit6s positives et la topologie de R. S. Martin," Ann. de !'Universitg de Grenoble, 23, 113-138 (1947-48). 3 Doob, J. L., "Discrete potential theory and boundaries," J. Math. and Mech., 8, 433-458 (1959). 4 G61, I. S., "Proximity relations and precompact structures," Proc. Kon. Ned. Akad. v. Weten- schappen, 62, 304-326 (1959). 6 (JG, I. S., Lecture notes on the foundations of analysis and geometry, Cornell University (1957-58). 6 Heins, M., "A lemma on positive harmonic functions," Ann. Math., 52, 568-573 (1950). 7 Martin, R. S., "Minimal positive harmonic functions," Trans. Am. Math. Soc., 49, 137-172 (1941). 8 Parreau, M., "Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann," Ann. Inst. Fourier, 3, 103-197 (1951).

ON CERTAIN HOMOGENEOUS COMPLEX MANIFOLDS BY PHILLIP A. GRIFFITHS DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY Communicated by D. C. Spencer March 7, 1962 The purpose of this note is to discuss some results on homogeneous vector bundles over homogeneous complex manifolds; complete proofs together with some more results and applications are to appear later. We refer to the papers of Wang6 and Bott3 for the terminology and basic results of the theory. Let, G, U be complex Lie groups such that X = G/U is a C-space and let p: U GL(EP) be a holomorphic representation of U; then we may form the homogeneous vector bundle E' - EP = G X uE - X. We recall that any C-space X fibers over a rational C-space X* with a complex a-torus as fiber: T2a -. X - X*. The maximal compact subgroup M of G acts transitively on X and transitively on the fibers of E'; letting EP be the sheaf associated to EP, H*(X, VP) is an M-module. We denote this induced representation by p*, p*:M GL(H*(X,U tP)); it is our problem to study the transformation p -_ p*. Let u = complex Lie algebra of U and hu = maximal abelian subalgebra of u; then hu C h, where h is a Cartan subalgebra of g = complex Lie algebra of G. If the weights of p on hu are the restrictions of weights on h to hu, we call p rational; otherwise p is irrational. If p-is rational, we may describe an element Jp giving the highest weight of an irreducible representation of M as follows: we assume that p is irreducible so that it is determined by its highest weight, again denoted by p. Then p is a weight on h which is dominant for u but may well not be dominant for Downloaded by guest on September 25, 2021