A (standard) Borel G-space is defined in connection with a standard Borel space which needs to be
specified first.
0.1 Basic definitions
- a. Standard Borel space.
Definition 0.1. A standard Borel space is defined as a measurable space, that is, a set
X equipped with a σ -algebra 𝒮, such that there exists a Polish topology on X with
S its σ-algebra of Borel sets.
- b. Borel G-space.
Definition 0.2. Let G be a Polish group and X a (standard) Borel space. An action
a of G on X is defined to be a Borel action if a : G × X → X is a Borel-measurable
map or a Borel function. In this case, a standard Borel space X 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 (Xn)n∈N 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.
