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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 and ℜ be algebras of subsets of a set Ω with ⊂ ℜ. Given a quasi-measure μ on , i.e., μ ∈ ba₊(), we denote by E(μ) the convex set of all quasi-measure extensions of μ to...

We show that the cardinality of a compact convex set W in a topological linear space X satisfies the condition that ${}^{\aleph ₀}=$. 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).

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.

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 ${}_{\nu }\left(X\right)$ and ${}_{\nu }\left(X\right)$ 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...

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We give some criteria for order boundedness of $E\left(\mu \right)$ in $ba\left(\Re \right)$, in the general case as well as for atomic $\mu$. Order boundedness implies weak compactness of $E\left(\mu \right)$. We show that the converse implication holds under some assumptions on $𝔐$, $\Re$ and $\mu$ or $\mu$ alone, but not in general.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We show that $E\left(\mu \right)$ is order bounded if and only if it is contained in a principal ideal in $ba\left(\Re \right)$ if and only if it is weakly compact and $extrE\left(\mu \right)$ is contained in a principal ideal in $ba\left(\Re \right)$. We also establish some criteria for the coincidence of the ideals, in $ba\left(\Re \right)$, generated by $E\left(\mu \right)$ and $extrE\left(\mu \right)$.

Let $K$ be a compact space and let $C\left(K\right)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C\left(K\right)$, including the following ones: (i) $\{x>0\}$ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. Moreover, the weak topology and that of pointwise convergence...

Let $X$ be a Banach lattice, and denote by $X_{+}$ its positive cone. The weak topology on $X_{+}$ is metrizable if and only if it coincides with the strong topology if and only if $X$ is Banach-lattice isomorphic to $l^1\left(\Gamma \right)$ for a set $\Gamma$. The weak${}^{*}$ topology on $X_{+}^{*}$ is metrizable if and only if $X$ is Banach-lattice isomorphic to a $C\left(K\right)$-space, where $K$ is a metrizable compact space.

CONTENTS1. Introduction..........................................................................................52. Vector measures with λ-everywhere infinite variation represented by series of simple measures.............113. Semicontinuity of some maps related to the variation map..................................................184. Sets of λ-continuous measures with (λ-) everywhere infinite variation.....................................235. Borel complexity of some spaces of vector measures........................................................266....

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 ${\prod }_{i\in I}{X}_i$ onto ${\prod }_{j\in J}{Y}_j$. We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_i$ onto $Y_{b\left(i\right)}$, i∈ I.