Definition 0.1
A standard Borel space is defined as a measurable space, that is, a set
equipped with a
-algebra
, such that there exists a Polish topology on
with
its
-algebra of Borel sets.
b. Borel G-space.
Definition 0.2
Let
be a Polish group and
a (standard) Borel space. An action
of
on
is
defined to be a Borel action if
is a Borel-measurable map or a
Borel function.
In this case, a standard Borel space
that is acted upon by a Polish group with a Borel action
is called a (standard) Borel G-space.
c. Borel morphisms.
Definition 0.3
Homomorphisms, embeddings or isomorphisms between standard Borel G-spaces
are called Borel if they are Borel-measurable.
Remark 0.1
Borel G-spaces have the nice property that the product and sum of a countable sequence of Borel G-spaces
are also Borel G-spaces. Furthermore, the subspace of a Borel G-space determined by an
invariant Borel set is also a Borel G-space.
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