Coordinatewise decomposition, Borel cohomology, and invariant measures

Benjamin D. Miller. "Coordinatewise decomposition, Borel cohomology, and invariant measures." Fundamenta Mathematicae 191.1 (2006): 81-94. <http://eudml.org/doc/282663>.

@article{BenjaminD2006,
abstract = {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.},
author = {Benjamin D. Miller},
journal = {Fundamenta Mathematicae},
keywords = {good sets; Glimm-Effros-style dichotomies},
language = {eng},
number = {1},
pages = {81-94},
title = {Coordinatewise decomposition, Borel cohomology, and invariant measures},
url = {http://eudml.org/doc/282663},
volume = {191},
year = {2006},
}

TY - JOUR
AU - Benjamin D. Miller
TI - Coordinatewise decomposition, Borel cohomology, and invariant measures
JO - Fundamenta Mathematicae
PY - 2006
VL - 191
IS - 1
SP - 81
EP - 94
AB - 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.
LA - eng
KW - good sets; Glimm-Effros-style dichotomies
UR - http://eudml.org/doc/282663
ER -