Abstract
Absolutely negligible sets in uncountable groups are considered in connection with the measure extension problem (for σ-finite invariant or quasi-invariant measures). In particular, it is proved that, for any uncountable solvable group (𝐺, ·), there exists a countable covering of 𝐺 consisting of 𝐺-absolutely negligible sets.
Let X be a topological space, that is, a space
with open sets such that the union of any collection of open sets is open and the
intersection of any finite number of open sets is open. A covering of
X is a collection of open sets whose union is
X. The covering is called countable if it consists
of a countable collection of open sets or finite if it consists of a finite
collection of open sets ; it is called locally finite if every point of
X is contained in some open set which meets only a
finite number of sets of the covering. A covering is called a
refinement of a covering U if every open set of X is
contained in some open set of . The space
X is called countably paracompact if every countable
covering has a locally finite refinement.
On Absolutely Negligible Sets in Uncountable Solvable Groups