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Compactness and extreme points of the set of quasi-measure extensions of a quasi-measure

100%
Lipecki Z.
Instytut Matematyczny Polskiej Akademii Nauk
EN
The memoir is based on a series of six papers by the author published over the years 1995-2007. It continues the work of D. Plachky (1970, 1976). It also owes some inspiration, among others, to papers by J. Łoś and E. Marczewski (1949), D. Bierlein and W. J. A. Stich (1989), D. Bogner and R. Denk (1994), and A. Ülger (1996). Let 𝔐 and ℜ be algebras of subsets of a set Ω with 𝔐 ⊂ ℜ. Given a quasi-measure μ on 𝔐, i.e., μ ∈ ba₊(𝔐), we denote by E(μ) the convex set of all quasi-measure extensions of μ to ℜ. Moreover, we denote by s, w and w* the strong, weak and weak* topologies of the dual Banach lattice ba(ℜ), respectively. Our starting point are the following two properties of E(μ) and extrE(μ), which are easy consequences of known results: (a) (E(μ),w*) is compact; (b) extrE(μ) is closed in (ba(ℜ),s). We study the following conditions related to (a) and (b): (i) (E(μ),s) is compact; (ii) (E(μ),w) is compact; (iii) s and w coincide on E(μ); (iv) s and w coincide on extrE(μ); (v) s and w* coincide on extrE(μ); (vi) w and w* coincide on extrE(μ); (vii) extrE(μ) is closed in (ba(ℜ),w); (viii) extrE(μ) is closed in (ba(ℜ),w*); (ix) (extrE(μ),s) is compact; (x) (extrE(μ),w) is compact; (xi) (extrE(μ),w*) is compact; (xii) (extrE(μ),s) is discrete; (xiii) (extrE(μ),w) is discrete; (xiv) (extrE(μ),w*) is discrete; (xv) extrE(μ) is dense in (E(μ),w); (xvi) extrE(μ) is dense in (E(μ),w*). In most cases, we find various equivalent conditions expressed in topological, affine-topological and measure-theoretic terms. To this end, we use, in particular, the antimonogenic component $μ^{a}$ of μ. (This is the minimal ν ∈ ba₊𝔐 such that ν ≤ μ and E(μ-ν) is a singleton.) Here are some sample results: (viii) holds if and only if $μ^{a}$ is atomic; both (xiii) and (xiv) are equivalent to the condition that $μ^{a}$ have finite range; (xvi) holds if and only if $μ^{a}$ is nonatomic. One of our main tools is an affine-topological representation of E(μ) for atomic μ as the countable Cartesian product of simplex like sets. We also study some other topological properties of extrE(μ), such as zero-dimensionality and various kinds of connectedness. Some of our results involve the cardinality 𝔪 of extrE(μ). In general, there are no restrictions on 𝔪 except for 𝔪 ≠ 0. However, if μ is nonatomic, then $𝔪^{ℵ₀} = 𝔪$. The case where 𝔪 ≤ ℵ₀ is also thoroughly investigated.
2
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Letter to the Editor

95%
Lipecki Z.
Studia Mathematica
|
2014
|
tom 224
|
nr 2
191
3
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Denseness and Borel complexity of some sets of vector measures

95%
Lipecki Z.
Studia Mathematica
|
2004
|
tom 165
|
nr 2
111-124
EN
Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $𝓔_{ν}(X)$ and $𝒜_{ν}(X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient conditions that $𝓔_{ν}(X)$ [resp., $𝒜_{ν}(X)$] be dense in ca(Σ,ν,X) [resp., ca(Σ,X)]. We also show that $𝓔_{ν}(X)$ and $𝒜_{ν}(X)$ are always $G_{δ}$-sets and establish necessary and sufficient conditions that they be $F_{σ}$-sets in the respective spaces.
4
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Semivariations of an additive function on a Boolean ring

95%
Lipecki Z.
Colloquium Mathematicum
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2009
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tom 117
|
nr 2
267-279
EN
With an additive function φ from a Boolean ring A into a normed space two positive functions on A, called semivariations of φ, are associated. We characterize those functions as submeasures with some additional properties in the general case as well as in the cases where φ is bounded or exhaustive.
5
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Cardinality of some convex sets and of their sets of extreme points

95%
Lipecki Z.
Colloquium Mathematicum
|
2011
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tom 123
|
nr 1
133-147
EN
We show that the cardinality 𝔫 of a compact convex set W in a topological linear space X satisfies the condition that $𝔫^{ℵ₀} = 𝔫$. We also establish some relations between the cardinality of W and that of extrW provided X is locally convex. Moreover, we deal with the cardinality of the convex set E(μ) of all quasi-measure extensions of a quasi-measure μ, defined on an algebra of sets, to a larger algebra of sets, and relate it to the cardinality of extrE(μ).
6
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On vector measures which have everywhere infinite variation or noncompact range

65%
Drewnowski L., Lipecki Z.
Instytut Matematyczny Polskiej Akademii Nauk
EN
Abstract Let X be an infinite-dimensional Banach space, let Σ be a σ-algebra of subsets of a set S, and denote by ca(Σ,X) the Banach space of X-valued measures on Σ equipped with the uniform norm. We say that a nonzero μ ∈ ca(Σ,X) is everywhere of infinite variation [has everywhere noncompact range] if |μ|(A) = ∞ or 0 [{μ(E): E ∈ Σ, E ⊂ A} is not relatively compact or equals {0}] for every A ∈ Σ. Let λ be a nonatomic probability measure on Σ, and denote by ca(Σ,λ,X) the closed subspace of ca(Σ,X) consisting of λ-continuous measures. Analogously as above, we define measures in ca(Σ,λ,X) that are λ-everywhere of infinite variation or have λ-everywhere noncompact range. Using the Dvoretzky-Rogers theorem, we give two constructions of an absolutely convergent series of λ-simple measures in ca(Σ,λ,X) such that the sum of each of its subseries is λ-everywhere of infinite variation. In particular, the normed space P(λ,X) of Pettis λ-integrable functions with values in X lacks property (K), and so is incomplete. These results refine and improve some earlier results of E. Thomas, and L. Janicka and N. J. Kalton. One of the constructions also yields the existence of an infinite-dimensional closed subspace in ca(Σ,λ,X) all of whose nonzero members are λ-everywhere of infinite variation. Moreover, modifying some ideas of R. Anantharaman and K. M. Garg, we prove that the measures that are λ-everywhere of infinite variation form a dense $G_δ$-set in ca(Σ,λ,X). From this we derive an analogous result on measures that are everywhere of infinite variation and the closed subspace of ca(Σ,X) consisting of nonatomic measures. Similar results concerning measures that have [λ-] everywhere noncompact range are also established. In this case, the existence of X-valued measures with noncompact range must, however, be postulated. We also prove that the measures of σ-finite variation form an $F_{σδ}$-, but not $F_σ$-, subset of ca(Σ,λ,X), and the same is true for P(λ,X) provided that X is separable. Finally, we consider the special case when X is a Banach lattice and, for X nonisomorphic to an AL-space, we note analogues of some of the results above for positive X-valued measures on Σ.
7
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Uniqueness of Cartesian Products of Compact Convex Sets

49%
Lipecki Z., Losert V., Spurný J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2011
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tom 59
|
nr 2
175-183
EN
Let $X_i$, i∈ I, and $Y_j$, j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of $∏_{i∈ I} X_i$ onto $∏_{j∈ J} Y_j$. We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_i$ onto $Y_{b(i)}$, i∈ I.
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