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Showing papers on "Bounded function published in 1971"


Journal ArticleDOI
TL;DR: In this paper, a class of symmetric relatively compact perturbations satisfying analyticity conditions with respect to the dilatation group was studied and the existence of an absolutely continuous part having spectrum [0, ∞] was proved.
Abstract: We study a class of symmetric relatively compact perturbations satisfying analyticity conditions with respect to the dilatation group inRn. Absence of continuous singular part for the Hamiltonians is proved together with the existence of an absolutely continuous part having spectrum [0, ∞). The point spectrum consists in R−{0} of finite multiplicity isolated energy bound-states standing in a bounded domain. Bound-state wave functions are analytic with respect to the dilatation group. Some properties of resonance poles are investigated.

1,049 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of estimating the state of a linear dynamic system using noise-corrupted observations, when input disturbances and observation errors are unknown except for the fact that they belong to given bounded sets, is considered.
Abstract: This paper is concerned with the problem of estimating the state of a linear dynamic system using noise-corrupted observations, when input disturbances and observation errors are unknown except for the fact that they belong to given bounded sets. The cases of both energy constraints and individual instantaneous constraints for the uncertain quantities are considereal. In the former case, the set of possible system states compatible with the observations received is shown to be an ellipsoid, and equations for its center and weighting matrix are given, while in the latter case, equations describing a bounding ellipsoid to the set of possible states are derived. All three problems of filtering, prediction, and smoothing are examined by relating them to standard tracking problems of optimal control theory. The resulting estimators are similar in structure and comparable in simplicity to the corresponding stochastic linear minimum-variance estimators, and it is shown that they provide distinct advantages over existing schemes for recursive estimation with a set-membership description of uncertainty.

698 citations


Journal ArticleDOI
TL;DR: In this article, a class of orthomodular lattices which admit no bounded measures was constructed, and the purpose of this paper is to construct a set of such lattices.

282 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that an in-stable diffeomorphism has only hyperbolic periodic points and if p is a periodic point of period k then the Arth roots of the eigenvalues of dff are bounded away from the unit circle.
Abstract: S Smale has recently given sufficient conditions for a diffeomorphism to be Q-stable and conjectured the converse of his theorem The purpose of this paper is to give some limited results in the direction of that converse We prove that an in- stable diffeomorphism / has only hyperbolic periodic points and moreover that if p is a periodic point of period k then the Arth roots of the eigenvalues of dff are bounded away from the unit circle Other results concern the necessity of transversal intersection of stable and unstable manifolds for an fi-stable diffeomorphism Introduction We will say that a diffeomorphism /: M—s- M of a compact manifold is Cl-stable if (a) there is a neighborhood N(f) of/in the C1 topology such that g e N(f) implies there is a homeomorphism « from the nonwandering set of/, Q(f) to the nonwandering set of g, 0(g) which satisfies g-« = «•/; and (b) if p is a periodic point off then dim Ws(p;f) = dim Ws(h(p); g) Property (b) is not usually included in the definition of Q-stable (see (3)), but it is a weak condition which is very natural and is apparently necessary for the proof of one of our lemmas (22) In his paper (4), S Smale provides sufficient conditions for a diffeomorphism to be £2-stable One of his conditions is that the nonwandering set have a hyperbolic structure Recall that a closed invariant set A is said to have a hyperbolic structure if (a) There is continuous splitting of the restriction of the tangent bundle to A,

280 citations


Book
01 Jan 1971
TL;DR: In this paper, Narasimhan presented the part of the theory of several complex variables pertaining to unramified domains over C. Topics discussed are Hartogs' theory, domains in holomorphy, and automorphism of bounded domains.
Abstract: Drawn from lectures given by Raghavan Narasimhan at the University of Geneva and the University of Chicago, this book presents the part of the theory of several complex variables pertaining to unramified domains over C . Topics discussed are Hartogs' theory, domains in holomorphy, and automorphism of bounded domains.

264 citations


Journal ArticleDOI
TL;DR: In this paper, a boundary is attached on which incomplete geodesics terminate as well as inextensible timelike curves of finite length and bounded acceleration, and the construction is free ofad hoc assumptions concerning the topology of the boundary and the identification of curves defining the same boundary point.
Abstract: To any space time a boundary is attached on which incomplete geodesics terminate as well as inextensible timelike curves of finite length and bounded acceleration. The construction is free ofad hoc assumptions concerning the topology of the boundary and the identification of curves defining the same boundary point. Moreover it is a direct generalization of the Cauchy completion of positive definite Riemannian spaces.

221 citations


Journal ArticleDOI
TL;DR: In this article, a general estimation theorem is given for a class of linear functionals on Sobolev spaces, which are those which annihilate certain classes of polynomials, and an interpolation scheme of Hermite type is defined in N-dimensions.
Abstract: A general estimation theorem is given for a class of linear functionals on Sobolev spaces. The functionals considered are those which annihilate certain classes of polynomials. An interpolation scheme of Hermite type is defined inN-dimensions and the accuracy in approximation is bounded by means of the above mentioned theorem. In one and two dimensions our schemes reduce to the usual ones, however our estimates in two dimensions are new in that they involve only the pure partial derivatives.

219 citations


Journal ArticleDOI
TL;DR: The novel criterion of maximizing the expected average compound return, which asymptotically leads to maximizing of geometric mean, is shown to be arbitrary.
Abstract: Because the outcomes of repeated investments or gambles involve products of variables, authorities have repeatedly been tempted to the belief that, in a long sequence, maximization of the expected value of terminal utility can be achieved or well-approximated by a strategy of maximizing at each stage the geometric mean of outcome (or its equivalent, the expected value of the logarithm of principal plus return). The law of large numbers or of the central limit theorem as applied to the logs can validate the conclusion that a maximum-geometric-mean strategy does indeed make it “virtually certain” that, in a “long” sequence, one will end with a higher terminal wealth and utility. However, this does not imply the false corollary that the geometric-mean strategy is optimal for any finite number of periods, however long, or that it becomes asymptotically a good approximation. As a trivial counter-example, it is shown that for utility proportional to xγ/γ, whenever γ ≠ 0, the geometric strategy is suboptimal for all T and never a good approximation. For utility bounded above, as when γ < 0, the same conclusion holds. If utility is bounded above and finite at zero wealth, no uniform strategy can be optimal, even though it can be that the best uniform strategy will be that of the maximum geometric mean. However, asymptotically the same level of utility can be reached by an infinity of nearby uniform strategies. The true optimum in the bounded case involves nonuniform strategies, usually being more risky than the geometric-mean maximizer's strategy at low wealths and less risky at high wealths. The novel criterion of maximizing the expected average compound return, which asymptotically leads to maximizing of geometric mean, is shown to be arbitrary.

202 citations


Journal ArticleDOI
TL;DR: In this article, a modified Marker and Cell computing method is presented for solving problems in incompressible hydrodynamics, which is applicable to time dependent problems in two spatial dimensions or three spatial dimensions with axial symmetry.

193 citations


Proceedings ArticleDOI
13 Oct 1971
TL;DR: In this paper, an algorithm for finding the biconnected components of an undirected graph and an improved version of an algorithm to find the strongly connected components of a directed graph are presented.
Abstract: The value of depth-first search or "backtracking" as a technique for solving graph problems is illustrated by two examples. An algorithm for finding the biconnected components of an undirected graph and an improved version of an algorithm for finding the strongly connected components of a directed graph are presented. The space and time requirements of both algorithms are bounded by k1V + k2E + k3 for some constants k1, k2, and k3, where V is the number of vertices and E is the number of edges of the graph being examined.

175 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the image of |z| < 1 under Vcffff k functions contains the disc of radius 1/k centered at the origin, and V>>\s k functions are continuous in |z |≦1 with the exception of at most [k/2+1] points on |z)|=1.
Abstract: Classes of functionsU k, which generalize starlike functions in the same manner that the classV k of functions with boundary rotation bounded bykπ generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior off α(z) = ∫ 0 z [f'(t)]α dt is determined for various classes of functions. It is shown that the image of |z|<1 underV kfunctions contains the disc of radius 1/k centered at the origin, andV k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1.

Journal ArticleDOI
TL;DR: In this article, the singular martingales on a von Neumann algebra with respect to a given ascending sequence of Von Neumann subalgebras as functionals on the C ∗ -algebra were studied.

01 Dec 1971
TL;DR: An algorithm is given for determining if two finite automata with start states are equivalent and the asymptotic running time of the algorithm is bounded by a constant times the product of the number of states of the larger automation with the size of the input alphabet.
Abstract: : An algorithm is given for determining if two finite automata with start states are equivalent. The asymptotic running time of the algorithm is bounded by a constant times the product of the number of states of the larger automation with the size of the input alphabet. (Author)

Journal ArticleDOI
TL;DR: In this paper, it is shown that if an autonomous linear randomly sampled system exhibits almost sure asymptotic stability, then the system is almost surely bounded input-bounded output.
Abstract: : The paper discusses the almost sure boundedness of linear and nonlinear randomly sampled systems. It is shown that if an autonomous linear randomly sampled system exhibits almost sure asymptotic stability, then the system is almost surely bounded input-bounded output. Moreover, for a bounded input, the second moment of the out-put remains bounded and this bound is easily computable. It is also found that linear or nonlinear systems which are almost surely asymptotically stable for a null input, remain almost surely bounded when the input consists of an uncorrelated noise with finite variance. (Author)

Journal ArticleDOI
TL;DR: In this article, it was shown that if certain conditions are satisfied there exists an unbounded continuum Λ of solutions (γ, u), withu(x)≥0, in the spaceRcffff 1×S HereS is a Banach space of real valued functions such as the one in this paper.
Abstract: Suppose Ω is a bounded region inR n In this paper we investigate the equationLu=λF(x, u),u|∂Ω=0 whereL is an elliptic partial differential operator and V 3F(x, t) is a positive function on ΩxR 1 which may have jump discontinuities with respect tot We show that if certain conditions are satisfied there exists an unbounded continuum Λ of solutions (γ, u), withu(x)≥0, in the spaceR 1×S HereS is a Banach space of real valued functions such as $$C\left( {\bar \Omega } \right) $$

Journal ArticleDOI
TL;DR: In this article, a solution of the boundary value problem is given, where Γ is a closed convex curve and Δu = −1 in the region D bounded by Γ.
Abstract: Let u be a solution of the following boundary-value problem: u¦Γ = 0, where Γ is a closed convex curve and Δu = −1 in the region D bounded by Γ. Then u has only one local maximum, and all its level curves are convex.

Journal ArticleDOI
TL;DR: It is shown that, under certain assumptions, problems of maximizing a sum of linear or concave-convex fractional functions on closed and bounded polyhedral sets can be transformed into equivalent ones of maximizing multiparameter linear or Concave functions subject to additional feasibility constraints.
Abstract: The paper deals with problems of maximizing a sum of linear or concave-convex fractional functions on closed and bounded polyhedral sets. It shows that, under certain assumptions, problems of this type can be transformed into equivalent ones of maximizing multiparameter linear or concave functions subject to additional feasibility constraints. The problems are transformed into those finding roots of monotone-decreasing convex functions. Where the objective function is separable, such a root is unique, and any local optimum is a global one, i.e., the objective function is quasi-concave. In problems involving separable linear fractional functions, under some additional assumptions, the parametric presentation results in a combinational property. Where the number of terms in the objective function is equal or less than three, this property leads to an efficient algorithm.

Journal ArticleDOI
TL;DR: In this article, the authors used complex analysis to obtain a simple direct treatment for the case of [a, b] where the crucial step is the construction of entire functions of exponential type which vanish at prescribed points not too close to the real axis.
Abstract: Let [a, b] be a bounded interval with a>O. Under what conditions on the sequence of exponents {A,,} can every function in LP[a, b] or C[a, b] be approxi mated arbitrarily closely by linear combinations of powers xAn? What is the distance between xA and the closed span Sc(xAn)? What is this closed span if not the whole space? Starting with the case of L2, C. H. Muntz and 0. Szasz considered the first two questions for the interval [0, 1]. L. Schwartz, J. A. Clarkson and P. Erdos, and the second author answered the third question for [0, 1] and also considered the interval [a, b]. For the case of [0, 1], L. Schwartz (and, earlier, in a limited way, T. Carleman) successfully used methods of complex and functional analysis, but until now the case of [a, b] had proved resistant to a direct approach of that kind. In the present paper complex analysis is used to obtain a simple direct treatment for the case of [a, b]. The crucial step is the construction of entire functions of exponential type which vanish at prescribed points not too close to the real axis and which, in a sense, are as small on both halves of the real axis as such functions can be. Under suitable conditions on the sequence of complex numbers {An} the construction leads readily to asymptotic lower bounds for the distances dk=d{xAk, Sc(xAn, nAk)}. These bounds are used to determine Sc(xAn) and to generalize a result for a boundary value problem for the heat equation obtained recently by V. J. Mizel and T. I. Seidman.

Journal ArticleDOI
TL;DR: In this paper, a backward error analysis of the diagonal pivoting method for solving symmetric systems of linear equations is presented, which shows that the elements of the associated error matrix can be bounded in time.
Abstract: A backwards error analysis of the diagonal pivoting method for solving symmetric (indefinite) systems of linear equations shows that the elements of the associated error matrix can be bounded in te...


Journal ArticleDOI
TL;DR: In this paper, the authors considered the convergence in distribution of the row sums of a triangular array of dependent random variables in the form of martingales within rows, and the results were obtained under conditions which parallel those of the classical case of convergence, to infinitely divisible laws with bounded variances, of the rows of elementary systems of independent random variables.
Abstract: Some results are obtained concerning the convergence in distribution of the row sums of a triangular array of certain dependent random variables. The form of dependence considered is that of martingales within rows, and the results are obtained under conditions which parallel those of the classical case of convergence in distribution, to infinitely divisible laws with bounded variances, of the row sums of elementary systems of independent random variables.

Journal ArticleDOI
TL;DR: In this paper, a classification of motion for Newtonian gravitational systems as time approaches infinity is presented, where the basic assumption is that the motion survives long enough to be studied, i.e., the solution exists in the interval (0, co). From this classification it is possible to obtain a sketch of the evolving Newtonian universe.
Abstract: In this paper we obtain a classification of motion for Newtonian gravitational systems as time approaches infinity. The basic assumption is that the motion survives long enough to be studied, i.e., the solution exists in the interval (0, co). From this classification it is possible to obtain a sketch of the evolving Newtonian universe. The mathematical study of Newtonian gravitational systems has a long history and has inspired a considerable amount of modern mathematics such as, among other topics, ergodic theory, algebraic topology, qualitative theory of differential equations and some functional analysis. Yet very little seems to be known about gravitational systems beyond the two-body problem. In 1922, J. Chazy [1] was able to classify the motion of the three-body problem as time, t, approaches infinity. In 1967, H. Pollard [7] obtained the first general n-body results as t -+ oo. He obtained results concerning the maximum and minimum spacing between particles as t -+ oo. His work suggests that the behavior of systems with nonnegative energy is in some sense a generalization of the twoand three-body problems. It is the purpose of this paper to sharpen these results and to provide the first classification of motion of the n-body problem, as t -+ oo independent of the sign of the energy. With this classification of motion a sketch of the evolution of Newton's universe as t -+ oo is possible. Also several remaining problems on the growth of systems are partially answered. It is interesting to note that previous classifications of motion have been attempted in terms of the sign of the total energy of the system. It turns out that this approach is far too restrictive and that the classification should be made according to the rate of separation of the particles, as is done here. It will be shown that in the absence of motion that we will call oscillatory and pulsating, the n-body problem is quite well behaved. It separates into clusters where the mutual distances between particles are bounded as t -+ oo. The clusters form subsystems characterized by the separation of clusters like t213. The centers of mass of the subsystems separate asymptotically from each other as Ct. Most of the results depend quite heavily on Tauberian theorems of the type of Landau [20, p. 194] (we follow the customary usage of the o-O symbols): Presented to the Society, January 23, 1970 under the title Classification of motion for gravitational systems; received by the editors March 12, 1970 and, in revised form, June 19, 1970. AMS 1969 suibject classifications. Primary 7034; Secondary 3440, 85XX.

Journal ArticleDOI
TL;DR: In this article, it was shown that a positive energy spectrum is not compatible with a decay law bounded by a decreasing exponential, and that the spectrum difficulty of wave functions for unstable, elementary particles was already recognized.
Abstract: We discuss the possibility of describing unstable systems, or dissipative systems in general, by vectors in a Hilbert space, evolving in time according to some non-unitary group or semigroup of translations. If the states of the unstable or dissipative system are embedded in a larger Hilbert space containing “decay products” as well, so that the time evolution of the system as a whole becomes unitary, we show that the infinitesimal generator necessarily has all energies from minus to plus infinity in its spectrum. This result supplements and extends the well-known fact that a positive energy spectrum is incompatible with a decay law bounded by a decreasing exponential. As an example of both facts, we discuss Zwanziger's irreducible, nonunitary representation of the Poincare group; and we find its minimal, unitary extension (the Sz.-Nagy construction). The answer provides a mathematically canonical approach to the Matthews-Salam theory of wave functions for unstable, elementary particles, where the spectrum difficulty was already recognized. We speculate on the possibility that the Matthews-Salam-Zwanziger representation might be a strong coupling approximation in the relativistic version of the Wigner-Weisskopf theory, but we have not shown the existence of a physically acceptable model where that is so.

Journal ArticleDOI
01 Feb 1971
TL;DR: In this paper, it was shown that if AB-BA is bounded and commutes with B, is | A \ B-A \ B\A\ necessarily bounded (i.e. whenever b' is essentially bounded).
Abstract: We show that it is possible for two selfadjoint opera- tors A and B in a Hilbert space H with bounded commutator AB—BA to have the property that |.4|.B — B\A\ is unbounded (where | A \ denotes the positive square root of A2). The proof re- duces to showing that for all natural numbers n, there exist a bounded positive operator U and a bounded operator V satisfying \\UV-VU\\^n\\UV+VU\\. Introduction. Interest in the above question arises from the fact that if H = L2( — oo, oo), Au =iu' and Bu = bu (where & is an a.e. differ- entiable function), then |^4JP —P|^4| is bounded whenever A B—BA is bounded (i.e. whenever b' is essentially bounded). Note that |^4|P—P|^4| is the singular integral operator (| AIE — P| ^41 )f(x) = 7T_1 p.v. f(x—y)~2(b(x)—b(y))f(y)dy. This is the one-dimensional L2 case of a more general theorem of Calderon (l). It was asked by T. Kato whether this case at least could be proved in an abstract setting, and in particular, whether |.4|P—P|^4| is bounded whenever AB—BA is bounded. Although we present two operators with AB—BA bounded and |j4|5— B\A\ unbounded, the question remains as to whether an abstract proof of Calderon's result can be found. In particular it is clear that in the special case (A =d/dx, B = b), AB — BA commutes with B. So it would be interesting to know the answer to the following question: If AB—BA is bounded and commutes with B, is | A \ B — B\A\ necessarily bounded? We comment further on this question at the end of the paper. Terminology. If A is a linear operator in a Hilbert space H, then D(A) denotes the domain of A. A linear manifold XED(A) is called a core oi A iiX is dense in D(A) under the norm ||«||;t = ||ii||24-||;lw||2. Throughout this paper the scalar field is assumed to be the field of complex numbers C. The result. (I) There exist two linear operators A and B in a

Journal ArticleDOI
TL;DR: In this paper, the authors studied weak solutions of elliptic equations of the form in a bounded domain, where it is assumed known about these solutions either that they are positive, or that estimates in certain norms hold for their negative parts.
Abstract: In this paper the authors study weak solutions of elliptic equations of the form in a bounded domain . It is assumed known about these solutions either that they are positive, or that estimates in certain norms hold for their negative parts. It is assumed moreover that an estimate on the -norm of the solution holds on some subdomain . Summability of such solutions with a weight function that vanishes at the boundary is established, and with the use of the results of Ja. A. Roĭtberg integral representations are given in terms of the Green's function for the Dirichlet problem.Bibliography: 8 titles.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every real-compact space with pseudocompact support has compact support and every i/i compact space is /?compact.
Abstract: Gillman and Jerison have shown that when A" is a realcompact space, every function in C(X) that belongs to all the free maximal ideals has compact support. A space with the latter property will be called fi-compact. In this paper we give several characterizations of /?-compact spaces and also introduce and study a related class of spaces, the ^-compact spaces ; these are spaces X with the property that every function in C(X) with pseudocompact support has compact support. It is shown that every realcompact space is ^-compact and every i/i-compact space is /?-compact. A family & of subsets of a space X is said to be stable if every function in C(X) is bounded on some member of #". We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space.

Journal ArticleDOI
TL;DR: In this article, the fixed membrane problem All + Xu = 0 in Q, u = 0 on aQ, for a bounded region Q of the plane, is approximated by a finite-difference scheme whose matrix is monotone.
Abstract: The fixed membrane problem All + Xu = 0 in Q, u = 0 on aQ, for a bounded region Q of the plane, is approximated by a finite-difference scheme whose matrix is monotone. By an extension of previous methods for schemes with matrices of positive type, 0(h 4) convergence is shown for the approximating eigenvalues and eigenfunctions, where h is the mesh width. An application to an approximation of the forced vibration problem Au + qu = J in Q, u = 0 in aQ, is also given.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the real parts of the functions in A(U) are uniformly dense in CR(∂U) if and only if each component of U is simply connected and each component is pointwise boundedly dense in H ∞(U).

Journal ArticleDOI
01 Feb 1971
TL;DR: A measure-preserving, invertible, ergodic automorphism of (X, iY, IA) has been defined in this article, where it is shown that the transformation in class S has a simple spectrum.
Abstract: A class of ergodic, measure-preserving, invertible point transformations is defined, called class S. Any measurepreserving point transformation induces a unitary operator on the Hilbert space of 22-functions. A theorem is proved here which implies that the operator induced by any transformation in class S has simple spectrum. [It is then a known result that the transformations in class S have zero entropy.] Let (X, 9, ,u) be a measure space, isomorphic to the unit interval with Lebesgue measure. A measurable, measure-preserving, invertible point transformation of X is called an automorphism of (X, i, u). A class of automorphisms, called class S for brevity, is defined below (Definition (4)). The purpose of this paper is to prove the following theorem: (1) THEOREM. Let r be an automorphism in class S. Then there exist arbitrarily small sets whose characteristic functions each generate ?2(dM) under the action of the unitary operator UT, where Ur is defined by UfQ(rx) =f (x). In particular U7 has simple spectrum. (2) DEFINITION. Let H be a Hilbert space, T a bounded normal operator on H. Let vE H. Let H(v) consist of the closure of the set of all elements of the form P(T, T*)v, where P(T, T*) denotes a polynomial in T and T*. To say that a vector vEH "generates H under the action of T" means that H= H(v). (3) DEFINITION. Let t = {IAt 1 < i ? m } be a finite, ordered collection of mutually disjoint measurable sets. Then t is called a partition. The union of the members of t need not be the whole space. Let ik be a sequence of partitions with the property that for every measurable set E, there exists a sequence of sets Ek such that each Ek is a union of members of (k, and pA(E A Ek)-*O as k-oo. Then it will be said that ik--e. Here e denotes the partition of the whole space into one-point sets. (4) DEFINITION. Let r be an automorphism of (X, iY, IA), {={A i1Ii

Journal ArticleDOI
TL;DR: In this article, the authors extend Bartle, Dunford, and Schwartz's theorem to the case where the control measure is finitely additive and strongly bounded (i.e., n(Ei)-+0 whenever {E t} is a disjoint sequence of sets).
Abstract: In this note we extend a theorem of Bartle, Dunford, and Schwartz [ l ] which states that for every countably additive measure denned on a (T-algebra there exists a positive \"control measure\" v such that v{E)—>0 if and only if ||/z||(.E) —»0, where ||/x|| is the semivariation of JJL. In this paper, ju, which is defined on a ring S, is assumed to be finitely additive and strongly bounded (^-bounded) [8] (that is n(Ei)-+0 whenever {E t} is a disjoint sequence of sets). The existing decomposition and extension theorems for vector measures can now be easily deduced by using the control measure. These applications will be presented in [2], 36 is a Banach space over the reals (the complex case is treated in a similar fashion); S* is the unit sphere in the conjugate space of 36. cr(g) denotes the cr-algebra generated by the class of sets 8. A ô-ring is a ring of sets closed under countable intersections.