Back to Fair Division Home
The monograph of Akemann and Anderson (1991) brings a generalization of Lyapunov's theorem to operator algebras. In the terminology of this paper, Lyapunov's theorem asserts that if Psi is a weak^* continuous linear map of an abelian non-atomic von Neumann algebra N to a complex vector space of finite dimension, and a\in (N +)1, the positive part of the unit ball of N , then there is an extreme point p of (N +)1 (i.e., a projection) such that Psi(p)=Psi(a). If X is a linear space containing a convex set Q and an affine map Psi of Q into a linear space Y, the main result of this paper gives conditions which ensure that Psi(E(Q))=Psi(Q), where E(Q) denotes the extreme points of Q. If Q is a compact subset of a locally convex space X such that facial dimension of Q is bigger than n and Psi is an continuous affine map of Q into Cn, then Psi(E(Q))=Psi(Q). If F is a weak^* closed subset of a non-atomic von Neumann algebra M and Psi is a weak^* continuous affine map of F into Cn, then Psi(E(Q))= Psi(Q). The authors are also dealing with atomic von Neumann algebras and prove a theorem which is very similar to a result of Elton and Hill (1987) . There is also a generalization of a result of Dvoretzky, Wald, Wolfowitz (1951) . The paper also contains many conjectures and many other small results.
Anantharaman (1977) gave a simple proof to a result of Diestel and Seifert (1976) that the range of a vector measure has the Banach-Saks property, based on the Banach-Saks theorem for Lp-spaces, 1<p<\infty. The author generalized the result to any measure with values in a quasi-complete locally convex space which is absolutely continuous with respect to a finite positive measure.
Anantharaman and Diestel (1991) gave some necessary and some sufficient conditions for a sequence (xn) in X(a Banach space with dual X^*) to lie in the closed convex hull of the range of an X-valued countably additive measure. The paper contains also several open problems. One of them has been already solved by Rodriquez-Piazza (1991) .
Anantharaman and Garg (1978) studied the properties of the range of a measure nu on a sigma-algebra taking values in a locally convex space for which there exists an equivalent non-negative measure. They considered conditions for the range of every restriction of nu to be convex. They characterized, as limits of zonohedra, weakly compact convex sets in a Banach space which are the range of a vector-valued measure.
Arbib (1966) simplified some of the proofs of Halkin (1964) and even improved the result. He assumed that a sub-algebra A of the class of Borel subsets B of [0,1] satisfies the following condition. For every epsilon>0 and A\in A , there exists an A -set Aepsilon\subseteqA with |mu(Aepsilon) - (1)/(2) mu(A)|< epsilon, where mu is Lebesgue measure on B . The author showed that if f is a finite-dimensional vector-valued integrable function over [0,1], then the set
Armstrong and Prikry (1981) showed first that the range of atomless, nonnegative and finitely additive vector measure defined on a Boolean algebra may not be closed. They also proved that the range of the vector measure as above, but defined on an F-algebra, is convex. This proof is based on Halmos' proof (1948) .
Armstrong and Prikry (1982) added a short note for their previous paper (1981) . The statement that the support of a Random measure on a quasi-F-space is Stonian is not correct. The authors gave a counterexample.
Artstein (1976) proved the following. Let f be a function from an interval [a,b] into a complete topological vector space X which is integrable in the since of Riemann-complete integration. Let RI(f) be the set of vectors in X that are the Riemann-complete-integral of f over a finite union of subintervals of [a,b]. Then RI(f) is a totally bounded set whose closure is convex. furthermore, if X has finite dimension then RI(f) is convex.
Artstein (1990) proved the Lyapunov convexity theorem using only Zorn's Lemma.
Avallone and Basile (1993) proved the convexity of the range of a closed-valued atomless finitely additive correspondence Phi defined on the measurable space (Omega, A ), where A is a sigma-algebra. The authors remarked that they use the sigma-algebra just for simplicity and it could be generalized for an F-algebra following steps as in Armstrong and Prikry (1981) .
Azarnia and Wright (1982) obtained results for operators on von Neumann algebras which, when specialized to commutative algebras, immediately give the theorem of Lyapunov (1940) .
Bartle, Dunford, and Schwartz (1965) showed that if E is a Banach space and A is a sigma-algebra, then the range of a vector-valued measure mu: A --E is weakly relatively compact.
Baumann (1966) extended a result of Borges (1965) about vector measures with infinite values to permit measures which are atomic.
Basile (1993) proved that the range of a closed-valued, finitely additive non-atomic set correspondence defined on a Boolean algebra has a convex closure.
Bazhenov (1985) proved the following theorems. A complete metrizable locally convex space F is a Lyapunov space if and only if F is finite-dimensional. Each almost finite-dimensional vector measure in a locally convex space is a Lyapunov measure. The article is written in Russian.
Bazhenov (1986) gave a complete characterization of locally convex spaces to which he extended the theorem of Lyapunov. He also showed that for an arbitrary Banach space X one can construct a measurable space such that every atomless vector measure, given on this measurable space, has a convex range as a subset of X.
Bazhenov (1986) wrote the following article in Russian. A locally convex Hausdorff space E is called Lyapunov space if each atomless countably additive vector measure on the Borel subsets of [0,1], taking its values in this space, has a convex range. The main theorem of this paper shows that the above defined space E is Lyapunov space if and only if a topology tau on Eis sequentially equivalent to a strong locally convex topology tau0 in E.
Bhaskara Rao, Candeloro, and Martellotti (1993) proved that any bounded finitely additive n-dimensional vector measure mu admits additive restriction, having the same range as mu. In addition, they proved that such a restriction can be chosen to be continuous when mu is continuous.
Blackwell (1951) proved the following extension of Lyapunov's theorem. Let mu1, ..., mun be atomless measures on a Borel sigma-algebra B of subsets of a space X and let A be any subset of n-dimensional Euclidean space. If f=(f1, ..., fn) is any B -measurable function defined on X with values in A, then the set { integral of f1 dm 1 ,…, fn dm n } is convex. Furthermore, if phiij, i=1,..., n, j=1,..., m, are B -measurable functions such that phiij is mui-integrable over X, and if D=(D1, ..., Dm) is a decomposition of X into m disjoint subsets, then the set { S j=1 to m ò Dj f 1jm 1,…, S j=1 to m ò Dj f njm n} is convex.
Blackwell (1951) also proved the following. Let mu=(mu1,...,mun) be a vector measure defined on (Omega, A ). Let f=(f1, ..., fn) be a measurable function defined on Omega with the range in a given bounded subset of Euclidean n-space. The author proved that the set {(ò f1dm 1,…, ò fndm n)} is closed and if mu is atomless, it is also convex. Part of the proof is based on the proof of Halmos (1948) .
Bolker's main result (1969) was to show that the following four properties of a convex subset Z of Rn are equivalent: (i) Z is the range of a vector measure. (ii) Z is a limit of sums of segments. (iii) The polar of Z is a central section of the unit ball of L1. (iv) The polar of Z is a projection body. A set Z which satisfies one of the above conditions (and hence all) is called a zonoid.
Bolker (1971) discussed in this short note what characterizations of a particular body can decide if the body is a zonoid.
Borges (1965) considered the following. Let mu1, ..., mun be signed atomless measures defined on a sigma-algebra of sets, and let E\subseteqRn be the set of finite values assumed by the vector-valued measure mu=(mu1, ..., mun). The author proved that if S is any support plane of E which meets the closure of E in a bounded set, then the intersection of S and the closure of E is contained in E. In particular, when mu is finite, this gives a new proof that E is closed.
Brams and Taylor (1995) presented a constructive procedure for dividing a cake among n people in the envy-free context. The central feature of their algorithm is that players trim pieces of the cake to create ties as for n<=3 known algorithms, but this time there are more pieces to be trimmed than there are players.
For given functions f1, ..., fk \in L1(Rn, Rm), weights p1, ..., pk: Rn --[0,1] with \sumpi =1, Bressan (1993) proved the existence of a partition {A1,..., Ak } of Rn such that the two functions
Brook and Graves (1980) gave a generalization of results of Knowles (1975) and of Tweddle (1972) to strongly countably additive map Phi defined on an algebra of subsets of a non-empty set X with values in a locally convex Hausdorff topological space over the scalar field of complex numbers.
Bryant proved (1975) the following theorem. Let B be the class of Borel subsets of [0,1], f be a vector valued integrable function, and mu be Lebesgue measure on B . Also, let A \subseteqB denote the class of subsets of [0,1] whose characteristic functions are continuous at all points of [0,1] with the exception of at most countable number of points. If D is any algebra of subsets of [0,1] such that A \subseteqD \subseteqB , then the set { ò A f dm : A Î D }
is convex. The proof requires only the methods of elementary analysis and linear algebra.Candeloro and Sacchatti-Martellotti (1979) proved the compactness and convexity of the range of an Rn-valued finitely additive continuous vector measure on a power set of an abstract space and of the weak compactness and convexity the weak closure of the range of such a Banach-space-valued measure. The paper is written in Italian with English summary.
Let X be a Banach space, A a sigma-algebra of sets and mu: A --X a countably additive measure; then the range of mu is always relatively weakly compact. Castillo and Sánchez (1994) proved that (i) if the closed unit ball of a (super-reflexive) space X lies inside the range of a measure then every bounded sequence (xn) in X admits a weakly 2-convergent subsequence; (ii) if X is reflexive and has an unconditional basis then every bounded sequence in X admits a weakly p^*-convergent subsequence, where p^* is the index conjugate to p.
For given a continuous map s--mus from a compact metric space into the space of atomless measures on (Omega, A ), Cellina, Colombo, and Fonda (1988) showed the existence of a family of measurable sets {Asalpha}alpha\in [0,1], increasing in alpha, such that for alphaÎ [0,1] mus(Asalpha)=alpha mus(Omega) and such that the map s --{Asalpha}alpha is continuous.
Cesari (1975) proved the following. Given any vector function f: [a,b] --Rn, whose components are L-integrable functions in [a,b], and any measurable set AÍ [a,b], then for every alphaÎ [0,1] there is a measurable subset BalphaÍ A with ò Ba f (t)dt = a ò A f (t)dt.
The author also proved, as a corollary, Lyapunov's theorem for the case of Lebesque measure on the real line. The approach relies only on elementary methods. Applications to linear control theory are given.
Chernoff (1951) proved that if muit, i=1,...,k, t=1,...,ni is a set of countably additive finite measures and if {E1,...,Ek} is the totality of decompositions of a space X into k pairwise disjoint measurable sets, the range of the vector psi with components muit(Ei), i=1,...,k, t=1,..., ni is bounded, closed, and in the atomless case also convex.
Crawford (1977) examined the game that arises when 2 players agree to use the divide-and-choose method as a resource allocation device.
Crawford and Heller (1979) considered the outcome of the divide-and-choose procedure for 2 players in the special case of one divisible and one indivisible good. They also considered a possible alternative to a modified classical method: allowing the divider to randomize the allocation of the goods in the bundle.
De-Lucia and Wright proved (1991) the following result. Let B be a Boolean sigma-algebra and let mu be a sigma-additive, G-valued measure on B , where G is an abelian topological group with a Hausdorff topology. If Z2is not a subgroup of G and if B satisfies the countable chain condition, then the range of mu is convex.
Diestel and Seifert (1976) proved the following. Let A be a sigma-algebra and let X be a Banach space. If mu: A --X is a measure, then every sequence in the range of mu has a subsequence whose arithmetic means rae norm-convergent. As a corollary they stated that the range of a vector measure has the Banach-Saks property.
Drewnowski and Lipecki (1995) wrote a 39-page long survey about vector measures with values in a real or complex Banach space and with relatively norm-compact ranges and with non-norm-compact ranges.
Drobot (1970) proved the following theorem. Let (Omega, A , mu) denote a measure space, where mu is a positive atomless measure with mu(Omega)=1, and let Hdenote a real Hilbert space. If f: Omega--H is an integrable function, then the closure of the set { ò A f dm : A Î A }
is convex. The proof is motivated by a method due to Halkin (1964) .Du's paper (1989) on Lyapunov's theorem in Hilbert spaces is in Chinese.
Dubins and Margolies (1980) proved two theorems which generalize the Lyapunov theorem. First, they showed that every invariant mean on an infinite amenable group has a convex range. Secondly, if G is an infinite accessible group, then its natural measure has also a convex range. At the end the authors remarked that an amenable group of sets is closed under disjoint unions, complements, and proper differences, but it is not closed necessary under intersections.
Dubins and Spanier (1961) proved the following. Let mu=(mu1, ..., mun) be a vector measure such that each mui is a countably additive finite real function on a sigma-algebra A of subsets of a set Omega. For fixed k, an ordered partition P=(A1, ..., Ak) of Omega, with Ai \in A , one gets a real matrix M(P)=(mui(Aj))ij. The main result is that the matrix range is compact. Moreover, if mu is atomless, then the matrix range is convex. Their proof is based on an idea of treating pure atomic and atomless cases separately and then using the decomposition of the vector measure.
Dvoretzky (1994) gave necessary and sufficient conditions for the range of a finite Rn-valued vector measure to be convex. A related result is given for Banach-valued set functions.
Dvoretzky, Wald, and Wolfowitz (1951) proved the following classical result. Let mu=(mu1,...,muk) be an atomless vector measure on (Omega, A ). Then for a natural number n=1 the set Rn(mu) of all vectors (mu(A1),..., mu(An)) \in Rnk, where (A1,..., An) is an ordered measurable partition of Omega, is a compact convex subset of Rnk.
Edwards (1987) extended the Dvoretzky-Wald-Wolfowitz theorem (1951) to appropriate vector measures with infinite-dimensional range space.
Elton and Hill (1987) proved that the distance from the convex hull of the range of an n-dimensional vector-valued measure is no more than (alphan)/(2), where alpha is the largest (one-dimensional) mass of the atoms of the measure.
Elton, Hill and Kertz (1986) proved the following. If mu1 ..., mun are atomless probability measures on the same measurable space, then there exists a measurable partition {Si}i=1n of Omega such that mui(Si)=(n+1-M)-1for all i=1, ..., n, where M is the smallest measure majorizing each mui. Moreover, these bounds are the best possible for the functional M. This result is an improvement of the Dubins-Spanier (1961) theorem and has a cake-cutting interpretation.
Even and Paz (1984) improved the fair division method of Banach and Knaster (in Steinhaus (1949) ), whose method called for up to n(n-1)/2 actual cuts. The authors offered a nondeterministic method for which the expected number of cuts is O(n).
Fedrizzi, Squillante and Ventre (1988) proved a one-dimensional Lyapunov theorem for \perp-decomposable measures.
Feller (1938) in his note showed that in general the range of finite atomless infinitely-dimensional vector measure may not be convex.
Ferakova and Nãther (1988) proved that the infinite-dimensional Lyapunov theorem for vector measures defined on a sigma-algebra is valid also in the case that the vector measures are defined only on a sigma-ring.
Fink (1964) gave a new solution to the fair division problem, i.e., how to divide a cake among n people so that each is assured a portion of the cake that he or she judges to be at least (1)/(n) of the cake. The solution consists of a finite (n(n-1))/(2) sequence of division problems each of which involves only two players.
Fischer and Schöler (1976) proved that the closure of the range of each atomless measure taking values in lp, 0< p <1 is a convex set.
Gardner (1978) discussed some partitions of given cakes with different shapes.
Gerencsér (1973) in this note gave another proof of the Lyapunov convexity theorem. He used the Hahn-decomposition theorem, Radon-Nikodym and the following theorem of Blackwell (1951) . Let the atomless probability measures mu1, ..., mun be defined on a sigma-algebra A of subsets of a set Omega. Then there exists a sub-algebra A 0 \subseteqA , containing Omega, on which the measures muiare still atomless and identical with each other. The article also contains interesting examples.
Glebov (1964) considered the following. Let mu be a measure on (X, A ) and let rj(t):X --Rn be bounded measurable functions for j=1, ..., k.
Let F (E1,…,Ek) = S j=1 to k ò Ej rj(t) dm
for disjoint measurable E1, ..., Ek. Then a bounded set Q \subseteqRn is the range of Phi (for some mu and r1, ..., rk) if and only if Qis the closure of the limit of a sequence of k-polyhedra. This article is written in Russian.Glicksberg (1983) gave more elementary proof of the Lyapunov convexity theorem than the one given by Lindenstrauss (1966) in the since that the Krein-Milman theorem is replaced by the existence of support functionals at boundary points of a convex set in Rn.
Glicksberg (1985) obtained an application of Lyapunov's theorem resulting from the use of support functionals which shows that the extreme points of the unit ball in L\infty under the usual norm yield the full image of the ball under finite dimensional continuous linear map.
Goller (1984) extended the convexity theorem of Lyapunov considering weak^*-continuous linear mappings of the dual space of a normed vector lattice into a topological Hausdorff space.
Gonzãlez-Velasco and Jones (1992) studied the range of an unbounded finite-dimensional vector-valued measure that is at least partly atomic. In the one-dimensional case they showed that if the range is dense in an interval [0,a] for some a>0 then it contains [0,a]. In the general case of arbitrary dimension d if e1, ..., ed are linearly independent vectors in Rd, then C denotes the convex cone. Now let mu be a measure such that any bounded subset of its range is in C, and such that the set of all mu(E) in C with E containing no atoms is bounded. The authors showed that if the range of mu is dense in (0,a]={ x\in rm interior C : 0<x <=a } for some a \in tex2htmldeferred interior C then it contains (0,a].
Gouweleew (1993) proved a generalization of the result of Dvoretzky, Wald, and Wolfowitz (1951) without the assumption that the measures involved in the theorem are atomless.
Gouweleeuw (1995) gave necessary and sufficient conditions on a finite non-negative vector measure mu=(mu1, ..., mun) defined on a sigma-algebra, under which the matrix range, the partition range and the range are convex. The paper also contains applications to partitioning and to classification problems.
Graves and Rüss' paper (1984) features strong and weak compactness in space of vector measures with relatively compact ranges in Banach spaces. The authors obtained necessary and sufficient conditions for compactness in certain spaces of vector measures and they added several applications of this result.
Halkin (1962) gave the necessary condition for the optimal control of a non-linear dynamical system. The result is a direct application of Lyapunov's convexity theorem.
Denote by mu Lebesgue measure on the Borel sigma-algebra B of [0,1]. An algebra A \subseteqB is called a continuous sub-algebra of B if for every A \in A there exists a class of sets { Dalpha: alpha\in [0,1] } \subsetA with D1=A, mu(Dalpha) =alpha mu(A) and Dalpha\subseteqDbeta if alpha< beta. Let F(A ) be the class of Lebesgue integrable functions from [0,1] into X (which is a finite dimensional Euclidean space). Halkin (1964) proved that the set { ò D f dm : D Î A }
is convex.Halkin (1965) proved, under the same assumptions as in Halkin (1964) with an extra condition that for any f Î F(A ) and pÎ X it is true that {tÎ [0,1]: <p, f(t) >>0} Î A , that the set { ò D f dm : D Î A } is closed and convex.
Halmos (1947) considered the following. Let E be a set and let A be a sigma-algebra of E. Let mu be a finite non-negative sigma-additive scalar measure on A . The author proved that the set of values of mu is closed. The author also wanted to prove that if mu and nu are measures in the above sense defined on A , then the range is a closed subset of the plane. However, the proof of this is not correct.
Halmos (1948) presented a simplified proof of Lyapunov's theorem, which has been used many times by other authors to derive extensions and generalizations of the theorem.
Hermes (1964) proved the following. Let X be a compact metric space and let yi, i=1,..., k, be continuous functions from X into En (Euclidean n-space). Let mu be a non-negative finite regular measure on the Borel subsets of X. For each xÎ X, let A(x) denote the finite set { y1(x), ..., yk(x) }, and co A(x) denote the convex hull of A(x). For each mu-measurable function z(x):X --En, let v(z) = ò X z(x) dm , let R(A) = {v(z): z(x) Î co A(x), all xÎ X}, and let R(co A) = {v(z): z(x) Î co A(x), all xÎ X}, Then R(A)=R(co A). The author applied his result to the theory of optimal control.
Herschbach (1996) generalized the classical theorem of Lyapunov to necessary and sufficient convexity conditions for the ranges of all finite-dimensional vector measures. In particular, he showed that the range of a vector measure mu defined on a sigma-algebra A is convex if and only if mu is atomless or the measure space (Omega, A , mu) possesses a characteristic atomic structure.
Hill (1983) showed the following. Suppose ncountries border on a region the ownership of which is in dispute. If each country's value of the territory is atomless, then there is a way of partitioning the disputed territory so each country receives a single piece adjacent to itself which it considers at least (1)/(n)the total value of the territory.
Hill (1985) proved the following. Let mu1, ..., mun be atomless finite measures. Then there exists a measurable partition A1, ..., An of X satisfying mui(Ai) =(1)/(n) h(|| mu1 ||, ..., || mun ||) for all i, where h(a1, ..., an) denotes the harmonic mean and where || mu||is the total variation of mu, and this bound is best possible.
Hill (1987) proved the following. Let mu1, ..., mun be probability measures on the same measurable space (Omega, A ). Then if all atoms of each mui have mess alpha or less, there is a measurable partition A1, ..., Anof Omega so that mui(Ai) =Vn(alpha) for all i, where Vn(.) is an explicitly given piecewise linear non-increasing continuous function on [0,1]. The paper contains applications to L1-spaces, to statistical decision theory and to the classical atomless case.
Hill (1988) studied a general class of measure-partitioning and proved that the critical case, where equality is attained, occurs when the measures are atomless and proportional. The author also gave many applications.
Hill (1993) wrote a survey about the topic of Lyapunov's theorem and its applications with a long list of reference. It also contains open problems in the area. However, since then some of them has been already solved.
Hoffman and Rothblum (1994) gave a simple proof of the Lyapunov convexity theorem using only a standard result about linear inequality systems and using Zorn's lemma.
Jamison (1974) gave a quick proof for a one-dimensional version of Lyapunov's theorem based on Zorn's Lemma.
Jones (1997) showed that, for dimension n=3 only, the direction set of all hyperplanes of mu-positive measure has Lebesgue measure zero on the unit sphere in R3, where mu is finite and atomless.
Kadets (1992) brought a generalization of Lyapunov's theorem for a single measure mu. It estimates the degree of non-convexity of the range of mu by the sizes of its atoms.
Kadets and Popov (1993) studied the case when the range of every measure has convex closure (in the infinite-dimensional case). They showed that for a Banach space X the following conditions are equivalent: (i) the range of every X-valued atomless measure of bounded variation has convex closure; (ii) L1 does not sign-embed in X; and (iii) each bounded operator from L1 into Xis norm-sign-preserving.
Kadets and Shekhtman (1992) proved that the closure of the range of each atomless measure taking values in lp, 1<=p < \infty with p \neq2, or in c0 is a convex set. The article is written in Russian.
Kellerer (1963) proved, relying on the non-constructive Borsuk-Ulam theorem, that the diagonal line segment is in the range of a finite atomless vector measure over the collection of all open sets. The article is written in German.
Kingman and Robertson (1968) gave necessary and sufficient conditions for the range of infinitely-many absolutely continuous measures to be convex and compact. They used an application of the Krein-Milman theorem to the space L\infty.
Kluvánek (1973) showed that if mu is an X-valued measure on a sigma-algebra, where X is a quasi-complete locally convex topological vector space, then the weak closure of the range of mu coincides with the closed convex hull of the range of mu when X is a metrazable or the underlying sigma-algebra is mu-essentially countably generated.
Kluvánek (1975) proved the following. Let Kbe convex, symmetric (about 0) and weakly compact. Let ||x'||K=\supx\in K|<x,x' >| be defined on the dual space. Then K is the closed convex hull of the range of a vector measure if and only if ||.||K is negative definite.
Kluvánek (1976) proved the following. Let Xbe a locally convex space with dual X' and let h(x) be the smallest linear lattice of functions containing X'. A conical measure on X is a non-negative linear functional on h(X). For a conical measure u and a point x in X, let x=r(u) if u(x')=<x', x > for all x' \in X'. Also for a conical measure u, let Ku= { r(v): v <=u }. A subset K of X is the closed convex hull of the range of a vector measure with range in X if and only if K=Ku for some conical measure u on X.
Kluvánek and Knowles (1974) gave the following generalization of Lyapunov's theorem. A vector measure mu: A --X is called Lyapunov vector measure if mu(A E)=mu({F \in A : F\subseteqE}) is convex and weakly compact for each E\in A . The authors showed that any closed vector measure is a direct sum of a Lyapunov measure and the extreme case of a non-Lyapunov measure. Also, they showed if mualpha: A alpha--X, alpha\in I, are vector measures and if mu: A --X is their direct sum, then each measure mualpha is Lyapunov if and only if mu is Lyapunov.
Kluvánek and Knowles (1975) wrote this book about vector measures.
Knowles (1975) gave necessary and sufficient conditions for a vector measure mu=mu1, mu2, ..., defined on a sigma-algebra of subsets of a time interval with values in a quasi-complete locally convex topological vector space, to be Lyapunov.
Koshi (1969-1970) gave a simple proof using only elemantary tools of measure theory to the following theorem: Let mu1, ...,mun be atomless completely additive measures on a sigma-complete Boolen lattice B and their total variations be finite. Then the range of mu=(mu1, ..., mun) is convex in the n-dimensional space.
Koshi and Lai (1981) presented two methods of proving the closedness of the range of any countable atomic finite measure on a sigma-algebra.
Kühn and Kühn (1998) showed that the direction set of all hyperplanes of mu-positive measure has Lebesgue measure zero on the unit sphere in Rn, where mu is finite and atomless. This result is a generalization of Corollary 1 in Jones (1997) and it has applications in fair division.
Kühn and Rösler (1998) showed that although convexity and compactness conclusions of Lyapunov's theorem may fail for measures defined on different sigma-algebras of the same set, they do hold if the sigma-algebras are nested, which is exactly the setting of classical optimal stopping theory.
LaSalle (1960) proved using directly the Lyapunov theorem the bang-bang principle, i.e., if a control system is being operated from a limited source of power and if one wishes to have the system change from one state to another in minimum time, then this can be done by at all times utilizing properly all of the power available.
Legut (1985) showed the existence of fair division for countably many participants and that the related cooperative game is totally balanced.
Legut (1987) showed the existence of epsilon-fair division for the game with continuum of players where epsilon is an arbitrary positive number.
Legut (1990) introduced a class of cooperative games arising from cooperation in a secondary division of an object Xamong n players, which is a special case of market games with a continuum of indivisible commodities coinciding with the class of totally balanced games. The author gave a characterization and some properties of the introduced class and at the end he gave two examples of games belonging to the defined class.
Legut and Wilczýnski (1988) defined a notion of an optimal partition of a measurable space into countably many sets according to given atomless probability measures. The authors showed that the set of optimal partitions is non-empty and gave an example related to statistical decision theory.
Lindenstrauss (1966) gave the shortest proof of Lyapunov's theorem based on the Radon-Nikodym, Banach-Alaoglu, and the Krein-Milman theorems.
Loeb (1973) showed that the range of a measure obtained by the addition of infinitesimal vectors is convex up to infinitesimal errors.
Lyapunov (1940) proved that the range of an atomless finite vector-valued measure is closed and convex.
Lyapunov (1946) showed by a counterexample that neither the convexity nor the closedness need to hold in the infinite dimensional case.
Maharam (1976) proved that the range of a finite-additive finite atomless measure defined on an F-algebra is closed and convex. The paper also contains interesting examples regarding finitely-additive measures.
Margolies (1978) proved that all left invariant means on infinite groups are atomless. This shows together with Theorem 2.2 in Armstrong and Prikry (1981) that the range {(mu1(A), ..., mud(A)) : A ranges over subsets of the group } of finitely many left invariant means (mu1, ..., mud) is convex.
Maritz (1980-81) showed that the bilinear integral of a set-valued function with values in an arbitrary Banach space is convex, provided the integral is a subset of a finite dimensional space and the measure is atomless. He also gave many interesting examples.
Martellotti (1985) showed the following. If mu is a continuous finitely additive measure defined on a sigma-algebra with values in a complete locally convex topological vector space with bounded range, then the weak closure of R(mu) is compact and convex.
Martellotti and Sambucini (1994) studied the relation between integral (representation in the form of an indefinite integral), differential (inequalities for increments) and topological (closedness of the range) conditions for finitely additive vector measures. They gave counterexamples that prove the impossibility of generalizing to vector measures some results for scalar measures.
Miheev (1975) established the convexity of the range of some classes of functions with values in a Banach space. These functions are defined and additive on some algebras of subsets of finite-dimensional spaces, but they do not need to be completely additive. The paper is written in Russian.
Mill and Ran (1996) gave a simpler proof of a result of Gouweleeuw (1995) .
Muscalu (1993) proved the following theorem. Let X be a finite-dimensional Banach space and mu: A --X be an atomless countably additive vector measure and nu: A --R_+ be a sigma-finite measure with mu << nu. Then for each alpha\in R_+ the set { mu(A) : A \in A , nu(A) <=alpha} is convex and norm compact.
Neyman (1981) proved that the following conditions on a zonoid Z, i.e., a range of an atomless vector measure, are equivalent: (i) the extreme set containing 0 in its relative interior is a parallelepiped; (ii) the zonoid Z determines the m-range of any atomless vector measure with range Z, where m-range of a vector measure mu is the set of m-tuples (mu(S1), ..., mu(Sm)) with disjoint measurable sets S1, ..., Sm; and (iii) there is a vector measure space (X, A , mu) such that any finite factorization of Z, Z=\sumZi, in the class of zonoids could be achieved by decomposing (X, A ). In the case of ranges of atomless probability measures (i) is automatically satisfied, so (ii) and (iii) hold.
The range of a countably additive atomless finite measure is by Lyapunov convex. Nunke and Savage (1952) showed a counter-example, expressed as a theorem, which shows that there is a finitely-additive atomless measure with a non-convex range. In the proof they used Zorn's Lemma.
Obha (1977) considered the following. Let A be a sigma-algebra of subsets of Omega, X a quasi-complete locally convex space and mu: A --X a vector measure. The vector measure mu is said to have the Bartle-Dunford-Schwartz property if there exists a finite non-negative measure nu on A such that mu(E) --0 if and only if nu(E) --0, E\in A . The paper contains a few sufficient conditions for this property. In particular, if mu is atomless and has this property, then the closed convex hull of the range of mu is equal to its weak closure.
Obha (1978) proved that the closure of the range of a measure of bounded variation on a sigma-algebra, with values in a Fréchet space with the Radom-Nikodym property, is compact and, if the measure has no atoms, also convex.
Olech (1968) proved the following theorem. If mu is an atomless finite-dimensional vector measure (not necessary finite), then the range of mu is convex, the closure of the range does not contain a line and each compact extreme face of the closure of the range is contained in the range. The article contains also examples showing that if mu is not finite, the range need not be closed.
Olech (1990) wrote a little survey about some proofs of Lyapunov's theorem and some known extensions.
Pineiro and Rodriquez-Piazza (1992) proved that the compact subsets of a Banach space X lie inside ranges of X-valued measures if and only if X^* can be embedded in an L1-space. They also proved if every compact subset of X lies inside the range of an X-valued measure of bounded variation, then X is finite dimensional.
Render and Strötmann (1996) proved that the range of a finite atomless vector measure mu over the collection of all open sets is convex. They used results from Kellerer (1963) , who proved that the diagonal line segment is in the range of mu over open sets.
Rickert (1967) proved the following. Let X be a measure space, and lambda a finite measure (countably additive) with values in Rn. The author introduced a certain positive measure nu, determined by the measure lambda, on the projective space Pn-1. Then the author showed that a necessary and sufficient condition that the range of lambda be a ball is that nu be an orthogonally invariant measure on Pn-1.
Rickert (1967) generalized his previous result (1967) to the following. Let X, lambda and nu be as in his previous paper. If X' is another measure space, with lambda' a finite measure on X' with values in Rn, and if nu' is the corresponding positive measure on Pn-1, then a necessary and sufficient condition that the range of lambda and the range of lambda' have the same convex hull is that (i) lambda(X)=lambda'(X') and (ii) nu=nu'. If only condition (ii) is satisfied, the range of lambda is a translate of the range of lambda'.
Rodriquez-Piazza (1991) answered an open problem of Anantharaman and of Diestel (1991) whether there can exist two vector measures with the same range, exactly one of them having bounded variation. The author proved that this is impossible, moreover, he proved that the range of a vector measure determines its total variation: measures with the same range have the same total variation.
Scozzafava (1978) proved the following. Let mu be a finitely additive probability measure on P (Omega), where Omega is an arbitrary infinite set. Then there always exists a sequence { Ek } of subsets of Omega with mu(Ek) >0 and Ei \capEj=\emptyset for i\neqj such that mu(\bigcupk=1\infty)=\sumk=1\inftymu(Ek). The fact that the range of an atomless finitely additive mu is the interval [0,1] is deduced as a corollary.
Sierpinski (1922) showed that the range of an atomless real-valued measure is an interval.
Steinhaus (1949) gave the rule for dividing an object fairly between n persons, n > 2. The solution is attributed to Banach and Knaster. The author extended the method to invisible objects, when each side payments are necessary.
Stromquist (1980) proved that a cake can be divided fairly among n people. He remarks that it can be done even if 'fair' means that all players must received their first choices.
Stromquist and Woodall (1985) showed that for given n atomless probability measures on the space I=[0,1] and a number alpha between 0 and 1, there exists a set K\subseteq[0,1], which is the union of at most n intervals, that has measure alpha in each measure. They also showed that if the underlying space is the circle S1 instead of I, then for all irrational and many rational values of alpha the set K is the union of at most n-1 intervals.
Suslov (1994) showed the following result. Let (Omega, A , mu) be a measure space with sigma-additive atomless measure mu: A --X of bounded variation and let R(E):={ mu(A): A \in A , A \subseteqE}. Also let the Banach space X be separable and possessing the Radon-Nikodym property. The contraction topology on a bounded closed convex subset A \subseteqXis defined to be the topology in which a neighborhood base of a point x \in A is constituted by the sets of the form x+ lambda(A-x), 0<lambda<=1. The author proved that if for every E \in A the set R(E) is dense in the closure of R(E) endowed with the contraction topology, then the set R(E) is closed in X for every E \in A .
Tang, Wan, Li and Zheng's paper The Lyapunov theorem in Hilbert space and its generalization from 1995 is in Chinese.
Tardella (1990) gave a new proof of the Lyapunov theorem based on the Shapley-Folkman theorem. It does not require any tools of functional analysis.
Turpin (1983) applied Lyapunov's theorem and its infinite-dimension generalization by Knowles to his study of atomless measures with values in a non-locally convex space. The author gave conditions under which the range of such a measure is contained in a convex bounded set or is itself convex and bounded. He also showed that every additive measure with values in a locally convex space is convexly bounded (i.e., the range is contained in a convex bounded set).
Tweddle (1972) considered the following. Let mu be a measure with values in E, a separated locally convex topological vector space. The author defined mu to be totally atomless if for every x' in E' the variation |x'.mu| is atomless. He showed that this notion is strictly stronger than merely being atomless, but that the two coincide when mu is of bound variation. Therefore, the following main theorem of the paper generalizes a theorem of Uhl (1969) : If mu is totally atomless, then the weak closure of the range of mu is convex and, if E is quasi-complete for the Mackey topology, also weakly compact.
Uhl (1969) showed that if X is a Banach space with the Radom-Nikodym property and if mu: A --X is a countably additive atomless vector measure of bounded variation, then the norm closure of the range of mu is convex and norm compact.
Urbansky (1976-77) improved result by Bartle, Dunford, and Schwartz (1965) by proving the following theorem. If E is quasi-complete and if mu: A --Eis a set-valued measure, where A is a sigma-algebra of subsets of a set S and if the values of mu are relatively compact, then so is its range.
Weller (1985) formalized the object (to be divided) as a measurable space (Omega, B ) and the tastes of the participating agents as finite atomless measures on this space. The author proved that a fair partition in (Omega, B ) exists and showed the relation of his result to the work of Dubins and Spanier (1961 .
Wnuk (1979) proved the following theorem. Let E be a metrizable complete topological linear space (i.e., an F-space). If every atomless measure defined on an arbitrary sigma-algebra, with values in E, has either convex or closed range, then E is finite dimensional. This shows that the Lyapunov's convexity theorem fails for every infinite-dimensional F-space.
Woodall (1980) proved the following. If each of n players defines a (suitable) measure on a compact convex cake I, then there exists a division of I into n connected parts, and an assignment of these n parts to the n people, in such a way that the piece of the cake assigned to each person is at least as large (in his/her own measure) as that assigned to anyone else. The proof uses Brouwer's fixed-point theorem and Hall's theorem on systems of distinct representatives.
Woodall (1986) proved constructively (without invoking the axiom of choice) that if mu1,..., mun are atomless probability measures not all the same, then there exists a measurable partition A1, ..., An such that mui(Ai) > (1)/(n) for each i.
Wu's 1988 paper is in Chinese.
Wulbert (1990) brought another proof of the classical Lyapunov's theorem using only Borsuk's antipodal mapping theorem.
Yaseen (1988) proved the following. If E is a locally convex space and mu: A --E is a vector measure of bounded variation such that mu has a Radon-Nikodym derivative, then the range of mu is conditionally compact in the topology of E. Moreover, if mu is atomless, then the closure of the range of mu is convex.
Yong (1984) , using the same tools as Lindenstrauss (1966) , proved Lyapunov theorem but without using induction (Lindenstrauss used induction in his proof).
Yorke (1971) gave another proof of the Lyapunov convexity theorem.