Let $C_g:C(\mathbb{R}^d;\mathbb{R}^d)\rightarrow C(\mathbb{R}^d;\mathbb{R}^d)$ be defined by $C_g(f)\triangleq f\circ g$ for some fixed $g \in C(\mathbb{R}^d;\mathbb{R}^d)$.

What are examples of Borel *probability* measures on $C(\mathbb{R}^d;\mathbb{R}^d)$ which are invariant with respect to $C_g$ and are non-atomic (preferably also locally positive: ie: positive value on non-empty open subsets).

Note: Here, $C(\mathbb{R}^d;\mathbb{R}^d)$ is equipped with the compact-open topology; ie: uniform convergence on compacts.