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Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications

Tim Austin (2010)

Fundamenta Mathematicae

We study a generalized notion of a homogeneous skew-product extension of a probability-preserving system in which the homogeneous space fibres are allowed to vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of 'direct integral' for a 'measurable family' of homogeneous spaces, which has a number of precedents in older literature. The main contribution of the present paper is the systematic development of a formalism for handling such extensions,...

Extensions of set functions.

Sergei V. Ovchinnikov, Jean Claude Falmagne (2003)

Mathware and Soft Computing

We establish a necessary and sufficient condition for a function defined on a subset of an algebra of sets to be extendable to a positive additive function on the algebra. It is algo shown that this condition is necessary and sufficient for a regular function defined on a regular subset of the Borel algebra of subsets of a given compact Hausdorff space to be extendable to a measure.

Extremal solutions of a general marginal problem

Petra Linhartová (1991)

Commentationes Mathematicae Universitatis Carolinae

The characterization of extremal points of the set of probability measures with given marginals is given in the general context of a marginal system. The sets of marginal uniqueness are studied and an example is added to illustrate the theory.

Extreme topological measures

S. V. Butler (2006)

Fundamenta Mathematicae

It has been an open question since 1997 whether, and under what assumptions on the underlying space, extreme topological measures are dense in the set of all topological measures on the space. The present paper answers this question. The main result implies that extreme topological measures are dense on a variety of spaces, including spheres, balls and projective planes.

F σ -mappings and the invariance of absolute Borel classes

Petr Holický, Jiří Spurný (2004)

Fundamenta Mathematicae

It is proved that F σ -mappings preserve absolute Borel classes, which improves results of R. W. Hansell, J. E. Jayne and C. A. Rogers. The proof is based on the fact that any F σ -mapping f: X → Y of an absolute Suslin metric space X onto an absolute Suslin metric space Y becomes a piecewise perfect mapping when restricted to a suitable F σ -set X ∞ ⊂ X satisfying f ( X ∞ ) = Y .

Factorization through Hilbert space and the dilation of L(X,Y)-valued measures

V. Mandrekar, P. Richard (1993)

Studia Mathematica

We present a general necessary and sufficient algebraic condition for the spectral dilation of a finitely additive L(X,Y)-valued measure of finite semivariation when X and Y are Banach spaces. Using our condition we derive the main results of Rosenberg, Makagon and Salehi, and Miamee without the assumption that X and/or Y are Hilbert spaces. In addition we relate the dilation problem to the problem of factoring a family of operators through a single Hilbert space.

Factors of ergodic group extensions of rotations

Jan Kwiatkowski (1992)

Studia Mathematica

Diagonal metric subgroups of the metric centralizer C ( T φ ) of group extensions are investigated. Any diagonal compact subgroup Z of C ( T φ ) is determined by a compact subgroup Y of a given metric compact abelian group X, by a family v y : y ∈ Y , of group automorphisms and by a measurable function f:X → G (G a metric compact abelian group). The group Z consists of the triples ( y , F y , v y ) , y ∈ Y, where F y ( x ) = v y ( f ( x ) ) - f ( x + y ) , x ∈ X.

Faithful zero-dimensional principal extensions

Tomasz Downarowicz, Dawid Huczek (2012)

Studia Mathematica

We prove that every topological dynamical system (X,T) has a faithful zero-dimensional principal extension, i.e. a zero-dimensional extension (Y,S) such that for every S-invariant measure ν on Y the conditional entropy h(ν | X) is zero, and, in addition, every invariant measure on X has exactly one preimage on Y. This is a strengthening of the authors' result in Acta Appl. Math. [to appear] (where the extension was principal, but not necessarily faithful).

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