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 <o uniformly continuous is cu = lub {y-1(V)t, where (p-l(l)), the inverse image of co under sp is determined by the structure 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 <o: X -* Y mapping the set X in the uniform space (Y, V)) and let cl = lub {so-lU(V)} Then a filter 1 in X is a Cauchy filter with respect to At if and only if <o(5) is a Cauchy filter in Y with respect to V) for every p e 4). Proof: Let ff be a filter in X such that so(ff) is a Cauchy filter for every (o 4). By the definition of cU for any given U E 'U, there exist functions (pi, ..., Sn e 4 and uniformities V1, ..., V. e VL such that (Pr-'(V1) n ... n Pn-'(Vn) C U. By hypothesis, there are suitable sets F1, . ., Fn in 9 such that (pi(Fi) X (p(Fi) C Vi (i = 1, ..., n). Therefore, Fj X Fj C spj-'(Vj) (i I1,.. n) and so F = F1 n ... nf Fn is an element of 3f which satisfies F X F C U. Since U e 'U is arbitrary, this shows that if is a Cauchy filter with respect to cU. In order to prove that so(5;) is a Cauchy filter for every (p E 4) when 5: is a Cauchy filter in X, it is sufficient to remark that each (p is uniformly continuous with respect to the structures 'U and 'C. For we have the following LEMMA 1. Let (X, 9U) and (Y, CL)) be uniform spaces and let <p: X -o Y be uni- formly continuous. Then so maps Cauchy filters into Cauchy filters. Proof: Let 9Y be a Cauchy filter with respect to 'l. A base for sp(F) consists of the sets p(F) (F e 5), so it is sufficient to prove that for each V in C) there is an F in 9: such that ip(F) X 4p(F) C V. Since (p is uniformly continuous V-'(u) < Ct and so U = '-'(V) e At. Thus, 9F being a Cauchy filter with respect to R, there is an F in 9F such that F X F C U. This set F satisfies the requirement. Next, we turn to the classical situations where the Martin boundary is already known and prove that the old and the new definitions coincide. The same reason- ing can be applied in the case of Euclidean domains, hyperbolic Riemann surfaces, and Green spaces and also for the so-called Martin exit boundary recently intro- duced by Doob in reference 3. The idea is very simple. First, we prove that a precompact uniform space (X, Ut) is uniquely determined by its Cauchy filters. This is the content of Proposition 3 below and it is proved by using a criterion for the possibility of a continuous extension of a map f: X -o Y from one uniform space (X, 'U) into another (Y, V)). As soon as the proposition is proved, we can deter- mine the Cauchy filters of the new uniform structure 91M by means of Proposition 1 and notice that these are identical with the Cauchy filters of the Martin compactifi- cation in the classical sense.