Let $(G,\circ)$ be a Polish group, with identity $e$.
Let $\Omega$ be the set of continuous functions $\omega:\mathbb{R} \to G$ such that $\omega(0)=e$.
For each $t \in \mathbb{R}$, define the projection $\pi_t:\Omega \to G$ by $\pi_t(\omega)=\omega(t)$.
Let $\mathcal{F}:=\sigma(\pi_t : t \in \mathbb{R})$. (Equivalently, $\mathcal{F}$ is the Borel $\sigma$-algebra generated by the compact-open topology on $\Omega$.)
For each $t \in \mathbb{R}$, define the $t$-shift map $\theta^t:\Omega \to \Omega$ by $\theta^t(\omega)(s)=\omega(s+t) \circ \omega(t)^{-1}$.
Let $\mathbb{P}$ be a probability measure on $(\Omega,\mathcal{F})$ that is ergodic with respect to the dynamical system $(\theta^t)_{t \geq 0}$. (The main case of interest to me is when $(G,\circ)=(\mathbb{R},+)$ and $\mathbb{P}$ is the "Wiener measure", meaning that the stochastic processes $(\pi_t)_{t \geq 0}$ and $(\pi_{-t})_{t \geq 0}$ are independent Wiener processes under $\mathbb{P}$.)
Does there exist a $\mathbb{P}$-null set $A \subset \Omega$ such that the set
$$\Omega_A \ := \ \{ \omega \in \Omega \, : \, \forall \ n \in \mathbb{N}, \ \exists \ t \geq n \ \ \textrm{s.t.} \ \theta^t(\omega) \in A \}$$
is not a $\mathbb{P}$-null set? Does there exist a $\mathbb{P}$-null set $A \subset \Omega$ such that $\Omega_A$ is a $\mathbb{P}$-full set, or even such that $\Omega_A$ is the whole of $\Omega$?
As I say, I am primarily interested in the case of the Wiener space described above. Many thanks.
