Showing papers in "Transactions of the American Mathematical Society in 1952"

Finitely additive measures

Abstract: 0. Introduction. The present paper is concerned with real-valued measures which enjoy the property of finite additivity but not necessarily the property of countable additivity. Our interest in such measures arose from two sources. First, the junior author has been concerned with the space of all finitely additive complex measures on a certain family of sets, which under certain conditions can be made into an algebra over the complex numbers. Second, both of us were informed by S. Kakutani of a result similar to our Theorem 1.23, the proof given by Kakutani being different from ours. The problem of characterizing finitely additive measures in some specific way occurred to us as being quite natural, and this problem we have succeeded in solving under fairly general conditions (Theorem 1.22). The body of the paper is divided into four sections. In §1, we consider finitely additive measures in a reasonably general context, obtaining first a characterization of such measures in terms of a countably additive part and a purely finitely additive part. Purely finitely additive measures are then characterized explicitly. In §2, we extend the theorem of Fichtenholz and Kantorovic [i](l) and thereby characterize the general bounded linear functional on the Banach space of bounded measurable functions on a general measurable space. In §3, we consider a number of phenomena which appear in the special case of the real number system. Here we exhibit a number of finitely additive measures which have undeniably curious properties. In §4, we describe connections between our finitely additive measures and certain countably additive Borel measures defined on a special class of compact Hausdorff spaces. We are indebted to Professor S. Kakutani for comments on and improvements in the results obtained. Throughout the present paper, the symbol R designates the real number system, and points are denoted by lower-case Latin letters, sets by capital Latin letters, families of sets by capital script letters. Sets of functions are denoted by capital German letters. For any set X and any A CZX, the characteristic function of A is denoted by \\a1. General finitely additive measures. 1.1 Definition. Let X be an abstract set, and let M be a family of subsets of X closed under the formation of finite unions and of complements. Let \"M be the smallest family of sets containing Vît and closed under the formation of countable unions and of complements.

A theory of spherical functions. I

Abstract: The fact that there exists a close connection between the classical theory of spherical harmonics and that of group representations was first established by E. Cartan and H. Weyl in well known papers; in fact, spherical harmonics arise in a natural way from a study of functions on G/K, where G is the orthogonal group in «-space and where K consists of those transformations in G under which a given vector is invariant. However, it is obvious that in order to get a theory applying to larger classes of special functions it is necessary to assume that only K is compact, and to study not only functions on G/K but also functions on G. The first nontrivial example was given by V. Bargman in 1947; in his paper [l] on the group G = SL(2, R), Bargman uses a maximal compact subgroup K of G (K is one-dimensional) and studies functions (g) which, for a given character x of K, satisfy 4>(kgk')=x(k)4>(g)x(k')In particular, such functions occur by considering finiteor infinite-dimensional irreducible representations of G; these representations have coefficients satisfying the above relation, and from explicit calculations it follows that these functions can be identified in a simple way with particular hypergeometric functions; the case where x(&) = 1 leads to Legendre functions of arbitrary index and to group-theoretical explanations of three important properties of these functions, namely, their differential equation, their representation by integral formulas, and their functional equation (which is connected with the classical addition formulas). At the same time I. Gelfand and M. Naimark were led to study similar questions by considering irreducible unitary representations of the complex unimodular group G = SL(n, C)\\ if K is a maximal compact subgroup of G, then it turns out that some of these representations have coefficients cj> satisfying (kgk') =(j}(g); these functions have properties entirely similar to those of the Legendre functions; however, they are not very interesting from the point of view of the theory of \"special\" functions because they can be expressed in closed form in terms of exponential functions (the formulas are very similar to H. Weyl's for the characters of compact groups). In 1950 very important results were published without proofs by I. Gelfand; in [6] Gelfand considers a (not necessarily compact) Lie group G and a compact subgroup K of G, and studies functions 0(g) satisfying (kgk')

A structure theory of Lie triple systems

Abstract: If in an associative algebra 2I a new composition is introduced by putting [ab] = ab ba, 2I becomes a nonassociative algebra 2%(L with a skew-symmetric product satisfying the Jacobi identity [[ab]c]+ [[bc]a]+ [[ca]b] =0. Furthermore, any abstract algebra having these properties is isomorphic to a subalgebra of some IL. These algebras, called Lie algebras, have been extensively studied, and in this paper we shall often refer to some of the many known results. If, on the other hand, we consider the system denoted by Sij comprising 2I and the product {ab} =ab+ba, we observe that 2T is a commutative, nonassociative algebra such that { { { aa } b } a } = { { aa }, { ba } }. Such algebras and their subalgebras are special Jordan algebras. This terminology is necessary since they are not characterized by the above properties. Abstract algebras defined by these properties are simply Jordan algebras. The starting point for our discussion is the observation that in an associative algebra [[ab]c] = { {cb}a} { {ca}b}, so that any special Jordan algebra may be treated as a subspace of an associative algebra closed under triple Lie products. Jacobson has characterized subspaces of Lie algebras closed under triple Lie products and called them Lie triple systems. Further, if in any Jordan algebra 2I we introduce the ternary composition [abc] = { { cb } a } { { ca} b }, 2I becomes a Lie triple system, the associator Lie triple system of 2I. Also, the mappings xRa = { xa } in a Jordan algebra constitute a Lie triple system which is a homomorph of the associator Lie triple system and is called the multiplication Lie triple system of the Jordan algebra. Suggested by these relations is the possibility of using Lie algebra methods and results in the study of Jordan algebras. Jacobson [7](3) has successfully done this, and it is not our present purpose to pursue further such applications of Lie triple systems. We shall, rather, regard such connections as motivation for the abstract study of the structure and classification of these systems. In section I we introduce some fundamental concepts and note some results of Jacobson concerning imbeddings of Lie triple systems in Lie algebras. Section II develops notions of the radical, semi-simplicity, and solvability as defined for Lie triple systems including proofs of the existence of a semi-simple subsystem complementary to the radical and of the decomposi-

On the application of the borel-cantelli lemma

Abstract: The celebrated Borel-Cantelli lemma asserts that (A) If ZP(Ek) < ?, then P (lim sup Ek) =0; (B) If the events Ek are independent and if ZP(Ek) = ?, then P(lim sup Ek)=l. In intuitive language P(lim stip Ek) is the probability that the events Ek occur "infinitely often" and will be denoted by P(Ek i.o.). This lemma is the basis of all theorems of the strong type in probability theory. Its application is made difficult by the assumption of independence in part (B). As Borel already noticed [1, p. 48 ff.], this assumption can be removed if we assume that(2)

Pairs of matrices with property l. ii(

Abstract: This note is concerned, for matrices with elements in an algebraically closed field of arbitrary characteristic p, with pencils generated by pairs of matrices with property L. A pair of n by n matrices is said to have property L if for a special ordering of the characteristic roots a< of A and Bi of B, the characteristic roots of \A+ptB are Xa<+p,/3,- for all values of X and pt. (See [1-5].) In §§1-5 another characterization of pairs of matrices with property L is given for a large class of such pairs. The method employed for this purpose is used in §6 for the study of pencils (not necessarily with property L) of diagonable matrices, i.e., matrices which are similar to a diagonal matrix. (These matrices are also called nondefective.) It is shown that for p=0, as well as for n^p, such pencils are always generated by commutative matrices. In §7 the significance of this result for general pencils of commutative matrices is investigated. 1. The ^-discriminant. The new characterization of pairs A, B of matrices with property L is obtained by considering those ratios X/p. for which \A +p,B has a multiple characteristic root. We see as follows that this is the case either for at most n(n — l) ratios or for every X/p.

Applications of martingale system theorems

Abstract: 1. Basic definitions. The following definitions are all relative to a triple (Q, 13, Pr) in which Q is a space with points denoted by w, 13 is a Borel field of subsets of Q which includes Q, and Pr is a completely additive set function defined on sets of 13 such that for AC3B, 0

First passage and recurrence distributions

Abstract: : A random walk on the integers is considered with transition probabilities p sub k for the transition k to k + 1 and 1 - p sub k for the transition k to k - 1. Distributions and moments are found for the length of time required to travel between any two given states. Some limiting results, applicable to more general results, are given.

Mappings between function spaces

Abstract: In the following we shall let U = [u] and Q3 = [v] denote arbitrary Banach spaces, and Z = [t] denote a Hausdorff space. X and 2) are to denote the spaces of all continuous functions mapping Z into U and into Q3 respectively. We shall let Q denote the space of all linear continuous mappings of U into Z3. A function K on Z to Q which is bounded on Z and continuous in the strong topology of Q induces a linear continuous operator K on I to 2) by the formula(2)

Banach spaces with the extension property

Abstract: Recently, in these Transactions, Nachbin [N] and, independently, Goodner [G] have shown that if B has the extension property and if its unit sphere has an extreme point, then B is equivalent to a function space of this sort; both authors have also proved that such a function space has the extension property The above theorem simply omits the extreme point hypothesis, and so establishes the equivalence My original proof, of which the proof given here is a distillate, depends on an idea of Jerison [j] Briefly, letting X be the weak* closure of the set of extreme points of the unit sphere of the adjoint B*, B can be shown equivalent to the space of all weak* continuous real functions / on X such that/(x) = —f( — x), and then properties of X are deduced which imply the theorem The same idea occurs implicitly in the proof below Note Goodner asks [G, p 107] if every Banach space having the extension property is equivalent to the conjugate of an abstract (L)-space It is known (this is not my contribution) that the Birkhoff-Ulam example ([B, p 186] or [HT, p 490]) answers this question in the negative, the pertinent Banach space being the bounded Borel functions on [0, 1 ] modulo those functions vanishing except on a set of the first category, with ||/|| = inf {K: \f(x) | g K save on a set of first category} 1 Preliminary definitions and remarks A point x is an extreme point of a convex subset K of a real linear space if x is not an interior point of any line segment contained in K (ie, if x=ty + (l — t)z, 0

An arithmetic theory of adjoint plane curves

Abstract: Introduction. In classical algebraic geometry the adjoint curves to an irreducible plane curve are an essential tool in the study of the geometry on the curve. In this paper we shall give an algebro-arithmetic development of the theory of adjoint curves, and shall extend the classical results to irreducible plane curves with arbitrary singularities defined over arbitrary ground fields. Our definition of the adjoint condition at a given singular point of the curve is stated in terms of the conductor between the local ring of the point and its integral closure. The fundamental properties of the adjoint curves are then derived from corresponding properties of the conductor. The single deepest and most important property of the adjoint curves is that, on a curve of order m, the adjoint curves of order m -3 cut out the complete canonical series. This property is equivalent to the fact that the degree of the fixed component of the adjoint series is twice the number of conditions which the adjoint curves impose on the curves of sufficiently high order(1). We shall give two distinct and independent proofs of this proposition. The first proof is a direct one, based upon a detailed analysis of the singularities of the given curve. This analysis, to which part I is devoted, applies equally well to algebraic number fields, and our treatment will include this case with that of algebraic function fields of one variable. The second proof is more indirect, depending upon the Riemann-Roch theorem and a generalization of the classical representation theorem of the differentials of the first kind. This proof, which will be given in part II, holds only for plane curves whose function field is separably generated over the ground field.

Primitive roots in a finite field

Abstract: with minimum m such that a(Qy) =0; y is said to belong to a(x). In particular if a(x) =Xpnx, Ore calls y a primitive root. It is easy to see that the two varieties of primitive roots are not identical; for example in the GF(24) defind by 04+0+1, 0 is a primitive root in the original sense but not in Ore's sense (since it belongs to x8+x4+x2+x). On the other hand it can be verified that 03 is primitive according to Ore but belongs to the numerical exponent 5. To avoid confusion we shall refer to ordinary primitive roots as primitive roots of the first kind, while those satisfying Ore's definition will be called roots of the second kind.-Ore proved that primitive roots of the second kind exist; indeed there are precisely b(Xn-1)CGF(pn), where 4 now denotes the Euler function for GF[p, x]. The equivalent result in terms of the existence of a normal basis (see ?2) had been proved by Hensel. It is natural to ask whether one can find a number 3EGF(pn) which is simultaneously a primitive root of both the first and second kinds. More generally if eI pn 1 and a(x) I Xpn_x, can one find a number ,B belonging to the numerical exponent e and the linear polynomial a(x)? We shall show that the first question is answered in the affirmative for pn sufficiently large; the second question also admits of an affirmative answer provided pn is large and e deg a(x) is sufficiently large. The method of proof is suggested by the proof of Vinogradoff's theorem that the least primitive root of a prime p is Q(pl/2+E); see [5, p. 178], also [3]. In the opposite direction we show (Theorem 4) that for given p, r there exist infinitely many irreducible polynomials P such that no polynomial of degree < r can be a primitive root of the second kind (mod P). Finally (Theorem 6) we obtain a bound for

On the structure of unitary groups

Abstract: 1. Let K be an arbitrary sfield with an involution J, that is, a one-to-one mapping £—*f of K onto itself, distinct from the identity, such that (£ + rj)J = ¡-J-\\-r]J, (¡-riY—i]J¿,J, and (¿/)/=?. Let E be an «-dimensional right vector space over K (n ^ 2) ; an hermitian (resp. skew-hermitian) form over £ is a mapping (x, y)—*f(x, y) of EXE into iC which, for any x, is linear in y, and such that/(y, x) = (f(x, y))J (resp./(y, x) = — (/(x, y))J). This implies that /(x, y) is additive in x and such that/(xA, y) =X//(x, y). The values/(x, x) are always symmetric (resp. skew-symmetric) elements of K, that is, elements a such that a/=o; (resp. aJ = —a). The orthogonality relation/(x, y) =0 relative to / is always symmetric. We shall always suppose that the form/is nondegenerate, or in other words that there is no vector in E other than 0 orthogonal to the whole space. Moreover, when the characteristic of K is 2, the distinction between hermitian and skew-hermitian forms disappears, and /(x, x) is symmetric for every xG-E; in that case we shall make the additional assumption that/(x, x) has always the form ¿--f-f (\"trace\" of £) for a convenient ¿£i?; this assumption is automatically verified when the restriction of / to the center Z of K is not the identity, but not necessarily in the other cases. A unitary transformation « of £ is a one-to-one linear mapping of E onto itself such that/(«(x), u{y)) =/(x, y) identically; these transformations constitute the unitary group U„(K, f). In a previous paper [5, pp. 63-82](1), I have studied the structure of that group in the two simplest cases, namely those in which K is commutative, or if is a reflexive sfield and the form / is hermitian; the present paper is devoted to the study of Un(K, f) in the general case. 2. We shall need the following lemma:

Representations of alternative algebras

Abstract: In this paper we apply to alternative algebras a definition of representation given by S. Eilenberg for nonassociative algebras satisfying multilinear identities. The corresponding alternative module generalizes the notion of a two-sided Sl-module as used in the study of associative algebras. Our chief object is to use the representation theory to obtain the generalization to alternative algebras of the theorem of A. Malcev on the strict conjugacy of semisimple components in Wedderburn decompositions. Since every alternative algebra gives rise to a (special) Jordan algebra, and every representation yields (Jordan) representations of this algebra, we can use recent results of N. Jacobson on representations of Jordan algebras. Doing this restricts our principal theorems to algebras of characteristic 0. Following certain preliminaries concerning derivations and associators, we prove the complete reducibility of representations of semisimple alternative algebras. We next prove the first Whitehead lemma for alternative algebras, generalizing G. Hochschild's result for associative algebras. This is sufficient to prove the Malcev theorem in case the square of the radical is {0 {. For all other types of algebras for which this theorem is known (Lie, associative, and Jordan), an inductive argument then suffices to complete the proof for an arbitrary radical. In the case of alternative algebras, however, without a stronger form of the Whitehead lemma a certain associativity condition would invalidate the inductive argument. Using the complete reducibility, we prove that this stronger form holds, and employ it in the proof of the Malcev theorem. In the concluding section we prove a generalization of a theorem due to Hochschild which, although independent of the representation theory, is related to our other results: an alternative algebra (of characteristic 0) is semisimple if and only if its derivation algebra is semisimple or {0}. We are indebted to Professor Jacobson for allowing us to see his paper, General representation theory of Jordan algebras, in manuscript form, and also for giving us valuable advice in connection with the proof of Theorem 2. 1. Representations and semidirect sums. A (nonassociative) algebra SI over a field F is called alternative in case

SOME BASIC THEOREMS IN DIFFERENTIAL ALGEBRA (CHARACTERISTIC p, ARBITRARY)

Abstract: cannot be taken over verbatim. This usually requires that the two cases be discussed separately, and this has been done below. The subject is treated ab initio, and one may consider that the proofs in the case of characteristic 0 are being offered for their simplicity. In the field theory, questions of separability are also considered, and the theorem of S. MacLane on separating transcendency bases [4] is established in the differential situation. 1. Definitions. By a differentiation over a ring R is meant a mapping u-*u' from R into itself satisfying the rules (uv)'=uv'+u'v and (u+v)' =u'+v'. A differential ring is the composite notion of a ring R and a differentiation over R: if the ring R becomes converted into a differential ring by means of a differentiation D, the differential ring will also be designated simply by R, since it will always be clear which differentiation is intended. If R is a differential ring and R is an integral domain or field, we speak of a differential integral domain or differential field respectively. An ideal A in a differential ring R is called a differential ideal if uGA implies u'GA. The ring { u+A } of residue classes of the differential ring R mod a differential ideal A is also a differential ring under the differentiation (u+A)'=u'+A. If F is a differential field, then from (v.u/v)'=v(u/v)'+v'(u/v) we obtain (u/v)'=(u'v-uv')/v2. If R is a differential integral domain, then its quotient field F becomes a differential field on setting (u/v) ' = (u'v - uv') /V2: it is this differential field which is intended when we speak of the

On the relation between Green’s functions and covariances of certain stochastic processes and its application to unbiased linear prediction

Abstract: As a result of the stimulus of the work of Wiener [1](2) and Kolmogoroff [2] the theory of linear prediction has developed during the past decade. In this paper we shall develop a method of solution for a class of problems which are natural generalizations of these that have been treated by the socalled auto-correlation theory as developed by Phillips and Weiss [3] or Cunningham and Hynd [4]. A basic difficulty in a linear prediction theory employing continuous observations appears to be that of solving the integral equations of the first kind that arise from the minimization of the error variance. Since, as in the auto-correlation theory, the prediction is based on a finite past history, such tools as Wiener's generalized Fourier analysis are not available and a general method of solution of the integral equations has not yet been devised. Here we develop a method of solution that requires hypotheses which are satisfied in many important physical applications. Moreover, as will be seen from Part I of the present paper, the kernel of our method is based on the intrinsically interesting relationship that exists between covariance functions of random processes generated by driving nth order linear differential equations by so-called "pure noise" and the Green's function of a suitably defined self-adjoint equation of order 2n. For physically stable linear differential equations with constant coefficients, it will be shown (Corollary 1.1) that such covariance functions are in fact Green's functions of a suitably defined self-adjoint problem. For linear differential equations with variable coefficients this is no longer true but such covariance

Galois theory of simple rings

Abstract: Introduction. The outer Galois theory, started by Jacobson [8], has been developed rather thoroughly [2; 3; 6; 12; 16]. The general Galois theory, dealing with general groups of automorphisms (with some restrictions though), has been established by Cartan [51 and Jacobson [9] in case of sfields. The purpose of the present paper is to offer a similar theory for simple rings with minimum condition('). The same has been given in fact in Hochschild [7] for simple algebras (finite over their centers). But the method breaks down in the case of general simple rings, infinite over their centers, and a new approach is necessary(2). The writer [14] has recently shown that if A is a simple ring and C is a weakly normal (cf. ?1 below) simple subring of A, then the A-left-, C-right-module A is fully reducible, and he has applied this fact, together with some methods in Dieudonne [6], to obtain a theorem of extension for isomorphisms in simple rings. It turns out that this full reducibility of A, with respect to the left-multiplication of A and the rightmultiplication of C, and some crossed product theorems, proved and used in the older papers by Azumaya and the writer [3; 12; 13; 16], are appropriate means for establishing the Galois theory(') for simple rings. In fact, if A is in particular a sfield, then A is clearly minimal (=irreducible) with respect to any operator domain containing the left-multiplication ring of A, and this fact underlies the Galois theory, as well as many other theories, for sfields. It is replaced, when A is a simple ring, by our A-C-full reducibility. The first section of the present paper gives some preliminary, though fundamental, lemmas on weakly normal simple subrings of a simple ring. In ?2 we introduce regular groups of automorphisms of a simple ring, which are the class of automorphism groups employed in our Galois theory, and consider their invariant systems. Conversely, we consider in ?3 the group of automorphisms leaving a subring, of a certain type, elementwise fixed. The Galois theory follows then in ?4. Although our method is rather different, we follow there the pattern of the algebra case in Hochschild [7 ]. Our theory can

The automorphism group of a lie group

Abstract: Introduction. The group A (G) of all continuous and open automorphisms of a locally compact topological group G may be regarded as a topological group, the topology being defined in the usual fashion from the compact and the open subsets of G (see ?1). In general, this topological structure of A (G) is somewhat pathological. For instance, if G is the discretely topologized additive group of an infinite-dimensional vector space over an arbitrary field, then A (G) already fails to be locally compact. On the other hand, if G is a connected Lie group, we shall show without any difficulty that the compact-open topology of A (G) coincides with the topology obtained by identifying A (G) with a closed subgroup of the linear group of automorphisms of the Lie algebra of G, as was done by Chevalley (in [1]) in order to make A (G) into a Lie group. We shall then deduce that A (G) is a Lie group whenever the group of its components, G/Go, is finitely generated('), where Go denotes the component of the identity element in G. The other questions with which we shall be concerned are the following: Let I(Go) denote the group of the inner automorphisms of Go, and let E(Go, G) denote the natural image in A (Go) of A (G). Regard I(Go) and E(Go, G) as

Homomorphisms of Jordan Rings of Self-Adjoint Elements

Abstract: In a previous paper [4](1) we have defined a special Jordan ring to be a a subset of an associative ring which is a subgroup of the additive group and which is closed under the compositions a→a 2and (a, b)→aba. Such systems are also closed under the compositions (a, b) → ab+ba= {a, b} and (a, b, c) → abc+cba. The simplest instances of special Jordan rings are the associative rings themselves. In our previous paper we studied the (Jordan) homomorphisms of these rings. These are the mappings J of associative rings such that $$ {\left( {a + b} \right)^J} = {a^J} + {b^J},\;{\left( {{a^2}} \right)^J} = {\left( {{a^J}} \right)^2},\;{\left( {aba} \right)^J} = {a^J}{b^J}{a^J} $$ (1) A second important class of special Jordan rings is obtained as follows. Let \( H \) be an associative ring with an involution a → a *, that is, a mapping a→a * such that $$ {\left( {a + b} \right)^*} = {a^*} + {b^*},\;{\left( {ab} \right)^*} = {b^*}{a^*},\;{a^{**}} = a $$ (2) Let \( H \) denote the set of self-adjoint elements h = h *. Then is a special Jordan ring. In this paper we shall study the homomorphisms of the rings of this type. It is noteworthy that the Jordan rings of this type include those of our former paper(2).