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Coordinatewise decomposition, Borel cohomology, and invariant measures

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Miller B.

Fundamenta Mathematicae

2006

tom 191

nr 1

81-94

Given Polish spaces X and Y and a Borel set S ⊆ X × Y with countable sections, we describe the circumstances under which a Borel function f: S → ℝ is of the form f(x,y) = u(x) + v(y), where u: X → ℝ and v: Y → ℝ are Borel. This turns out to be a special case of the problem of determining whether a real-valued Borel cocycle on a countable Borel equivalence relation is a coboundary. We use several Glimm-Effros style dichotomies to give a solution to this problem in terms of certain σ-finite measures on the underlying space. The main new technical ingredient is a characterization of the existence of type III measures of a given cocycle.

Coordinatewise decomposition of group-valued Borel functions

100%

Miller B.

Fundamenta Mathematicae

2007

tom 196

nr 2

119-126

Answering a question of Kłopotowski, Nadkarni, Sarbadhikari, and Srivastava, we characterize the Borel sets S ⊆ X × Y with the property that every Borel function f: S → ℂ is of the form f(x,y) = u(x) + v(y), where u: X → ℂ and v: Y → ℂ are Borel.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Miller B.

1 2007

1 2006

Wyniki wyszukiwania

Coordinatewise decomposition, Borel cohomology, and invariant measures

Coordinatewise decomposition of group-valued Borel functions