Range Of Vector Measure Research Articles

A note on the range of vector measures

Abstract We give another proof for Kluvanek and Knowles’ characterization of Liapounoff measures [KLUVANEK, I.—KNOWLES, G.: Vector Measures and Control Systems. North-Holland Mathematics Studies 20, Amsterdam, 1976] and of the fact that the range of an exhaustive measure with values in a complete locally convex space is relatively weakly compact.

Sequential Analogues of the Lyapunov and Krein–Milman Theorems in Fréchet Spaces

In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).

Anti-Compacts and Their Applications to Analogs of Lyapunov and Lebesgue Theorems in Frechét Spaces

We introduce anti-compact sets (anti-compacts) in Frechet spaces. We thoroughly investigate the properties of anti-compacts and the scale of Banach spaces generated by anti-compacts. Special attention is paid to systems of anti-compact ellipsoids in Hilbert spaces. The existence of a system of anti-compacts is proved for any separable Frechet space E. Using the constructed theory, we obtain analogs of the Lyapunov theorem on the convexity and compactness of the range of vector measures in the class of separable Frechet spaces: We prove the convexity and compactness of the range of vector measure in a space \( {E}_{\overline{C}} \) generated by an anti-compact \( \overline{C} \). Also, the nondifferentiability problem with respect to the upper limit is investigated for the Pettis integral. We obtain differentiability conditions for the indefinite Pettis integrals in terms of the new weak integral boundedness and the σ-compact measurability. We prove an analog of the Lebesgue theorem on the differentiability of the indefinite Pettis integral for any strongly measurable integrand.

Set-valued functions, Lebesgue extensions and saturated probability spaces

Recent advances in the theory of distributions of set-valued functions have been shaped by counterexamples which hinge on the non-existence of measurable selections with requisite properties. These examples, all based on the Lebesgue interval, and initially circumvented by Sun in the context of Loeb spaces, have now led Keisler and Sun (KS) to establish a comprehensive theory of the distributions of set-valued functions on saturated probability spaces (introduced by Hoover and Keisler). In contrast, we show that a countably-generated extension of the Lebesgue interval suffices for an explicit resolution of these examples; and furthermore, that it does not contradict the KS necessity results. We draw the fuller implications of our theorems for integration of set-valued functions, for Lyapunov's result on the range of vector measures and for the theory of large non-anonymous games.

Absolutely $p$-Summable Sequences in Banach Spaces and Range of Vector Measures

Absolutely $p$-Summable Sequences in Banach Spaces and Range of Vector Measures

Generalizations of the time optimal problem and Lyapunov theorem on the range of vector measures

Given an n × r integrable matrix function Y ( t ) , we extend the Lyapunov–Lindenstrauss theorem describing extreme points of the set { ∫ 0 T Y ( t ) u ( t ) dt | u ∈ I } from the Cartesian product I of r Lipschitz classes to the Cartesian product I = H ω [ 0 , T ] := H ω 1 [ 0 , T ] × ⋯ × H ω r [ 0 , T ] of classes H ω [ 0 , T ] of functions with the modulus of continuity majorized by the given concave modulus of continuity ω . We also explain the intimate relationship between the aforementioned problem and the characterization of extremal functions in the classical time minimization problem of optimal control T → inf ; x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) , u ( · ) ∈ H ω [ 0 , T ] , x ( 0 ) , u ( 0 ) = 0 , x ( T ) , u ( T ) = ( Λ ^ , Γ ^ ) , for locally integrable n × n - and n × r -matrix valued functions A ( t ) and B ( t ) , the collection ω = ( ω 1 , … , ω r ) of concave moduli of continuity, and Λ ^ ∈ R n , Γ ^ ∈ R r . Relying on these results, we solve the classical rendezvous problem of finding the optimal trajectory in the phase space ( x , x ˙ , x ¨ , … , x ( r ) ) , x ( r ) ∈ H ω ( R + ) , connecting two given points in R r + 1 . Then, we describe the extreme points of the set S ω , r , τ , a := { ( x ( τ ) , x ′ ( τ ) , … , x ( r ) ( τ ) ) | x ( r ) ∈ H ω [ 0 , T ] : x ( i ) ( 0 ) = a i , i = 0 , … , r } for a = ( a 0 , … , a r ) ∈ R r + 1 , τ > 0 . This problem is related to the Kolmogorov problem for intermediate derivatives where the triples ( x ( τ ) , x ( m ) ( τ ) , x ( r ) ( τ ) ) are considered for 0 < m < r .

The Range of Vector Measures over Topological Sets and a Unified Liapunov Theorem

Let (X, Σ, μ) be a finite, nonatomic, measure space. Let G=span{g1, g2, …, gn}⊆L1, and let the support of G be X, a.e. For f∈L∞, put M(f)=(∫Xfg1dμ, ∫Xfg2dμ, …, ∫Xfgndμ). Then Q={M(f):f∈B(L∞)} is a compact convex set. Liapunov's classsical theorem is that also Q={M(s):s∈extB(L∞)}. This paper characterizes when the functions, s, in Liapunov's theorem can further be restricted to being the signs of continuous functions. That is, suppose there is a topology on X, and that Σ is the Baire sets. Let S be the collection of supports of continuous non-negative functions. Theorem.The following are equivalent: (i)Q={M(s):s∈extB(L∞),ands=sgnf for some f∈C(X)}, (ii)for every g∈G, {x:g(x)>0}and{x:g(x)⩽0} are in S, a.e. The theorem is proved in a general setting. If the σ-field in the general theorem is arbitrary, the theorem becomes Liapunov's Theorem. The theorem above results when the σ-field is the Baire sets. A setting with the Borel sets produces Q as the range of M over extreme functions that are both lower semi-continuous a.e. and upper semi-continuous a.e.

Remarks on the range of a vector measure

A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p &gt; 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p &lt; ∞, no weakly-p-summable sub-sequences.

Zum satz von ljapunov in vektorverbänden

Let K be a closed convex order-bounded set of an order-complete vector lattice and let A be a him continuous linear operator. Then the equality AK = A(ext K) is proved, where ext K is the set of all extremal points of K. It is shown that various generalizations of the Ljapunov-theorem on the range of vector-measures are special cases of this general statement.

The range of vector measures into Orlicz spaces

The range of vector measures into Orlicz spaces

On a generalization of a theorem of Lyapunov

A constructive and simple proof is given of a generalization of a result of Lyapunov on the convexity of the range of vector measure.