Nonatomic Measure Research Articles

On the non-compactness of the core of nonatomic convexσ-continuous measure games

This paper concerns with the core of nonatomic games of form f(μ), whereμ is a nonatomic nonnegative measure and f is a continuous convex function on the domain ofμ. The main result of this paper is that the core of the game is not compact under the norm topology unless the game itself is a measure. This shows the largeness of the core in a sense other than that defined by Sharky for finite cases.

Orlicz spaces which areL p -spaces

Let Ф denote a finite-valued Orlicz function vanishing only at zero and consider the Orlicz spaceL Ф over a non-atomic, σ-finite and infinite measure space with the norm ∥ · ∥ being either the Luxemburg norm or (in the case where $$\mathop {\lim }\limits_{u \to 0} \Phi (u)/(u) = 0$$ and $$\mathop {\lim }\limits_{u \to + \infty } \Phi (u)/(u) = + \infty $$ the Orlicz norm. Ifp ∈ [1, + ∞) and $$||1_A + 1_B ||^P = ||1_A ||^P + ||1_B ||^P $$ for all disjoint setsA andB of positive and finite measure, thenL Ф is the Lebesgue spaceL p .

Fredholm weighted composition operators

We characterize the Fredholm weighted composition operators onC(X). In particular, ifX is a set with some regular property like intervals or balls inR n , our characterization implies that a weighted composition operator is Fredholm if and only if it is invertible. This equivalence is true for weighted composition operators onL p (μ), where μ is a nonatomic measure (1≤p<∞).

The value of certain pNA games through infinite-dimensional Banach spaces

LetB([0, 1]) be the Borel sets of [0, 1] and letX be an infinite-dimensional Banach space. Let μ:B([0, 1])→X be a countably-additive vector measure, and letf:X→R be a function withf (0)=0. LetpNA be the Banach space spanned by polynomials of nonatomic realvalued measures. Sufficient conditions are obtained for the gamefo μ to be inpNA, and a formula is developed for the value of such games. Moreover, examples are given to illustrate why the seemingly obvious finite-dimensional approximations are not applicable.

Orlicz Spaces Without Strongly Extreme Points and Without H-Points

AbstractW. Kurc [5] has proved that in the unit sphere of Orlicz space LΦ(μ) generated by an Orlicz function Φ satisfying the suitable Δ2-condition and equipped with the Luxemburg norm every extreme point is strongly extreme. In this paper it is proved in the case of a nonatomic measure μ that the unit sphere of the Orlicz space LΦ(μ) generated by an Orlicz function Φ which does not satisfy the suitable Δ2-condition and equipped with the Luxemburg norm has no strongly extreme point and no H-point.

On the Lyapunov convexity theorem with appications to sign-embeddings

It is proved (Theorem 1) that for a Banach space X the following assertions are equivalent: (1) the range of every X- valued σ- additive nonatomic measure of finite variation possesses a convex closure; (2) L1 does not signembed in X.

Extreme and Exposed Points in Orlicz Spaces

AbstractExtreme points of the unit sphere in any Orlicz space over a measure space that contains no atoms of infinite measure are characterized. In the case of a finite-valued Orlicz function and a nonatomic measure space, exposed points of the unit sphere in these spaces are characterized too. Some corollaries and examples are also given.

Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation

In this paper several monotonicity properties of the Luxemburg norm in Musielak-Orlicz spaces L φ ( μ) and E φ ( μ) over nonatomic measure spaces are characterized in terms of the function φ. For L φ ( μ) it is proved that all these properties coincide with the absolute continuity of the norm and φ > 0. Some applications to best approximation are given, even for general Banach lattices.

On the fair division of a heterogeneous commodity

Consider a heterogeneous but divisible commodity, bundles of which are represented by the (measurable) subsets of the good. One such commodity might be land. The mathematics literature has considered agents with utilities that are nonatomic measures over the commodity (and hence are additive). The existence of ‘α-fair’ allocations, in which each agent receives a utility proportional to his utility of the endowment of the entire economy, was demonstrated there. Here we extend these existence results to α-fair efficient allocations, envy-free allocations, envy-free efficient allocations, group envy-free and nicely shaped allocations of these types. We examine utilities that are not additive and relate the mathematics literature to the economics literature. We find sufficient conditions for the existence of egalitarian-equivalent efficient allocations. Finally, we consider the problem of allocating a time interval (uses of a facility). Existence of an envy-free allocation had been demonstrated in earlier literature. We show that any envy-free allocation is efficient as well as group envy-free. We extend this last result to a more general setting.

Shift operators on Banach spaces

J. R. Holub has obtained several results for shift operators on C( X). In this paper we answer some questions of Holub and obtain extensions of many of his results. In particular, we show that C( X, R) does not admit a shift operator if X has only countably many components and each component is infinite. We show that C( X, C) does not admit a shift operator for certain special compact Hausdorff spaces X. We show that there exists a compact Hausdorff space X which is not totally disconnected and both C( X, C) and C( X, R) admit shift operators. If 1 ⩽ p < ∞ and ( X, Σ, μ) is a σ-finite non-atomic measure space then L P R ( μ) does not admit a disjointness preserving shift operator. We also show that l P for 1 ⩽ p ⩽ ∞ is the only L P R ( μ) space which admits a disjointness preserving shift operator.

Operators preserving disjointness on rearrangement invariant spaces

Let X and Y be two rearrangement invariant spaces on a measure space (Ω, Σ, μ) with a finite, nonatomic measure μ . We show that if there exists a non-zero order continuous disjointness preserving operator T: X —• Y, then X C.Y. This result has many consequences. For example, if T: LP(Ω, Σ, μ) -> Lq(Ω, Σ, μ) (0 < p < q < oo) preserves disjointness, then T = 0. 1. Notation and preliminary facts. Recall that a (linear) operator T: X —• Y between vector lattices is said to be a disjointness preserving operator if |JCI| Λ \x2 = 0 in X implies \Tx{ Λ \Tx2 = 0 in Y. All vector lattices are assumed to be Archimedean, and all operators on normed or linear metric spaces are assumed to be continuous. Let (Ω, Σ, μ) be a measure space with a finite σ-additive nonatomic measure and S(Ω,Σ,μ) be the space of all (equivalence classes of) measurable real valued functions. Throughout the work we will use the representation of the space S as the space Coo(Q) of all continuous extended functions on the Stone space Q of S. (See [10] for details.) We retain the same notation μ for the corresponding measure on Q, which is defined on the σ-algebra ΣQ consisting of all subsets of the form (E\N) U (N\E), where E is a clopen (closed and open) subset of Q and N is a first category subset of Q. It is well known that μ(D) = 0 if and only if D is a nowhere dense subset of Q. (Any extremally disconnected space Q with such a measure is sometimes called a hyperstonian space.) A subspace X of 5(Ω, Σ, μ) is called a rearrangement invariant (r.i.) ideal if

THE ABSENCE OF NONTRIVIAL BOUNDEDLY COMPACT TCHEBYCHEFF SETS IN THE SPACES $ L_\varphi$

It is proved that if is a nonatomic measure space and an even function nondecreasing on and such that , 0$ SRC=http://ej.iop.org/images/0025-5734/69/2/A07/tex_sm_2114_img6.gif/> for 0$ SRC=http://ej.iop.org/images/0025-5734/69/2/A07/tex_sm_2114_img7.gif/>, and for all , 0$ SRC=http://ej.iop.org/images/0025-5734/69/2/A07/tex_sm_2114_img10.gif/>, then the space does not contain boundedly compact Tchebycheff sets with more than one point.

Large symmetric games are characterized by completeness of the desirability relation

The paper presents a characterization of continuous cooperative games (set functions) which are monotonic functions of countably additive non-atomic measures. The characterization is done through a natural desirablity relation defined on the set of coalitions of players. A coalition S is at least as desirable as a coalition T (with respect to a given game υ (in colational form)), if for each coalition U that is disjoint from S ∪ T, υ( S ∪ U) ⩾ υ( T ∪ U). The characterization asserts, that a game υ is of the form υ = f · μ, where μ is a non-atomic signed measure and f is a monotonic and continuous function on the range of μ, if, and only if, it is in pNA′ (i.e., it is a uniform limit of polynomials in non-atomic measures or equivalently it is uniformly continuous function in the NA-topology) and has a complete desirability relation.

Properties of measures on totally ordered spaces

The aim of this paper is to study the properties of measures on a totally ordered spaceX. It is shown that if the cardinal of every closed discrete subset ofX is of measure zero then every non-atomic measure onX is τ-additive. It is also proved that a continuous Borel measure μ onX is Radon if and only if μ is perfect, τ-additive and the support of μ is almost a totally ordered space.

Strong singularity, disjointness, and strong finite additivity of finitely additive measures

The most striking difference between finitely additive measures and countably additive measures is that the Hahn decomposition theorem may be only approximate if countable additivity fails. A corollary is that whereas two singular countably additive measures are disjoint in giving full measure to disjoint sets, this is usually only approximately true without countable additivity. One natural question is “Which finitely additive measures are disjoint from all singular measures?” In the Stone space setting, the question becomes “Which Radon measures have support disjoint from all singular Radon measures?” If so, the support is a P 1-set of Atalla which is more general than a P-set. Construction of such measures is considered at length. Existence of nonmolecular nonatomic measures based on P 1-points with support a P 1-set is equivalent to the existence of a P 1-homeomorph of β N in the Stone space. If ( X, Σ, μ) is a positive localizable measure space, then BA( Σ) has L ∞∗(X, Σ, μ) as a band and and L 1( X, Σ, μ) as a subband. It is shown that the band complementary to L ∞∗(X, Σ, μ) consists of measures disjoint to μ so supp(μ) is a P 1-set in the Stone space of Σ. A notion of strong singularity to μ intermediate to disjointness and singularity is shown to be equivalent to strong finite additivity (admitting countable partitions by null sets) for elements of L ∞∗(X, Σ, μ) . The band of L ∞∗(X, Σ, μ) orthogonal to L 1( X, Σ, μ) consists of strongly finitely additive measures precisely when the measure algebra Σ μ satisfies the λ-chain condition for a nonreal valued measurable cardinal λ or when the maximal ideal space of L ∞( X, Σ, μ) does not have real valued measurable cellularity. A μ in BA( Σ) has a countably additive ideal of negligible sets iff it is not strongly finitely additive on any set of positive measure. On the Stone space, this corresponds to Radon measures with P-sets as supports. Even for purely finitely additive measures of this type, much of countable additive measure theory holds, for null functions are precisely those with null support. It is not known for Σ = 2 x whether purely finitely additive measures of this type exist. Their existence may be independent of ZFC and/or other axioms such as Martin's, real-valued measurable cardinal, or measurable cardinal. A locally compact space Y is sham-compact so that C( X) = C b ( X) = C c ( X) precisely when its remainder in any compactification is a P-set. In this case M( X) = M b ( x) = M c ( X). The latter strong of identities is the definition of sham-sham compact spaces. These are characterized by having a P 1-set as remainder in any compactification. New examples of sham-compact open subsets of the maximal ideal space Z μ of a positive localizable measure space ( X, Σ, μ) are given based on Maharam's homogeneity character. However, no open subset of Z μ can be sham-sham compact without being sham compact. This is true when Z μ is replaced by any compact space which locally the support of a Radon measure.

Une caracterisation du type de la loi de Cauchy-conforme sur ℝ n

The conformal Cauchy law is the probability on ℝn with densityC/(1+‖X‖2)n. It is shown that for a non-atomic measure μ on ℝn the following is true: its type is invariant under inversions of ℝn if and only if it is the type of a conformal Cauchy law. (The type of a measure is defined as the set of its images under similarities and translations.)

Dynamics induced on the ends of non-compact manifold

AbstractLet ℋ denote the group of homeomorphisms of a σ-compact manifold M which preserve a locally positive non-atomic measure μ. Such a manifold can be compactified by adjoining ideal points called ‘ends’, collectively denoted by E. Every homeomorphism h in ℋ induces a measure preserving system (E, , μ*, h*) where is the algebra of clopen subsets of E, μ* is a 0-∞ measure induced on E by μ, and h*: E → E is a μ*-preserving homeomorphism. For any induced homeomorphism σ = f*, where f belongs to ℋ, define ℋσ = {h ∈ ℋ: h* = σ}. We prove that ergodicity is generic in ℋσ for the compact-open topology if and only if (E, , μ*, σ) is incompressible and ergodic. Furthermore ℋσ contains an ergodic homeomorphism if and only if (E, , μ*, σ) is incompressible. Since the identity on M induces the identity on E, which is incompressible, our results establish that every manifold (M, μ) supports an ergodic μ-preserving homeomorphism.

A continuous version of Liapunov’s convexity theorem

Given a continuous map s ↦ μs, from a compact metric space into the space of nonatomic measures on T, we show the existence of a family (Aαs)α∈[0,1], increasing in α and continuous in s, such thatμs(Aαs)=αμs(T)(α∈[0,1]).

Measures admitting extremal extensions

Introduction. We consider the abstract measure extension problem: Let 91 c ~B be a-algebras of subsets of some set f2; for a given measure # on 91, what can be said about the (convex) set M ( ~ ; #) of measures on ~3 extending #? It is wellknown that M (~B; #) may be empty, or non-empty but without extremal points, or consist of exactly one measure (cf. examples in [2], [1 1]). In particular, any information about the existence of extremal extensions seems to be valuable, not only because most of the concrete constructions known in the literature lead to extremal extensions, but also because, for a nonatomic measure # on 91, in order to "define" S f d# for some ~3-measurable bounded function f as the set {Sf dr: v ~ M (~B, #)}, it often suffices to consider extremal extensions of # (cf. [12]). In this paper we use the technique of dilation developed in [5] to describe extremal extensions through the notion of 91-atoms of ~3, introduced in [6] in the context of entropy. F rom this description we obtain criteria for the existence of extremal extensions (Theorems 1 and 2) and show that the set of measures admitt ing extremal extensions forms a monotonical ly closed hereditary subcone, thereby generating a projection band in the lattice of signed measures (Theorem 3). The paper concludes with necessary and sufficient conditions for the existence of simultaneous extremal extensions of a given family of measures (Theorem 4).

Daugavet's Equation and Operators on L 1 (μ)

Generalizing a result of Babenko and Pichugov, it is shown that if T is a weakly compact operator on L1 (It), where 1A is a a-finite nonatomic measure, then III + TIl = 1 + 1ITIh. A characterization of all operators T on L' (p) having this property is also given. In [2] Daugavet proved that if T is a compact operator on C(,i), then I + TiI = 1 + 11ThI, while Babenko and Pichugov [1] subsequently showed the same is true for a compact operator on L'(IL) (see [61 for recent extensions). In general, if X is a Banach space and T E L(X), then T is said to satisfy Daugavet's equation [4] if III TiI = 1 + 1.ITIh. This property of an operator arises naturally in the consideration of problems of best approximation in function spaces where it has been utilized by a number of authors (e.g. [4], and the references cited in [1]). In a recent paper [5] it was shown that Daugavet's equation actually holds for any weakly compact operator on C(,u). The purpose of this note is to give an analogous extension of the theorem of Babenko and Pichugov to weakly compact operators by proving that if , is a a-finite nonatomic measure on a space S, and T is a weakly compact operator on L1(I), then IlI + TII = 1 + IITII (Theorem 1). Related results concerning extensions and generalizations of this theorem, including a characterization of all operators T on L1 (it) for which IIITII = 1+ lT II (Theorem 2) are also given. We begin with a simple result concerning the norming of operators on Ll (,u). As usual we denote by L(X) the space of all bounded linear operators on a Banach space X, by XA the characteristic function of a measurable set A c S, and by IIf! II and IgIk0, the norms of functions f E L1(,V) and g E L??(,i), respectively. The complement of a set A in S is denoted by A'. PROPOSITION 1. Let T E L(L1('i)). Then given any E > 0 there is a measurable set A C S for which 0 ITIh E. PROOF. Since it is a-finite, (Ll(IL))* = L??(,t). Therefore if T E LZ(L'(,it)), then T* E L(LOO (,i)) and IITII = IIT*II = sup11 g 11= =T* g I I Hence given any e > 0 there is a function g E L??(I,) for which higilgo = 1 and IIT*glloo > ITIh e/2. Since IIT*g I I = ess supt I(T*g)(t)I it follows that there is a measurable set A C S for which m(A) > 0 and I(T*g)(t)l > 11TIh E for all t E S. Replacing A (if necessary) by a subset B C A for which 0 < m(B) < E and on which the sign of (T*g)(t) is constant (which may be done since it is nonatomic), and replacing g by (-g) if Received by the editors April 7, 1986. 1980 Mathematics Sject Clssfication (1985 Revision). Primary 47B38, 47A30; Secondary 46E15.