A standard Borel space is defined as a measurable space, that is, a set $X$ equipped with a $\sigma$ -algebra $\mathcal{S}$, such that there exists a Polish topology on $X$ with $S$ its $\sigma$-algebra of Borel sets.

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 $a:G\times X\to X$ 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.

Homomorphisms, embeddings or isomorphisms between standard Borel G-spaces are called Borel if they are Borel–measurable.