Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu := \mu \circ g^{-1}$ for all $g \in G$. Ergodicity means that $$\mbox{if $\mu(A \triangle g^{-1}A) = 0$ for all $g \in G$, then $\mu(A) = 0$ or $\mu(A^c) = 0$,}$$ where $A \triangle g^{-1}A$ denotes the symmetric difference of $A$ and $g^{-1}A$.
Fix some $p \in [0,\infty]$, and let $L^p := L^p(X,\mathbb R; \mu)$ denote the space of $p$-integrable functions. Let $\mathcal B^p := \mathcal B(L^p)$ denote the Borel $\sigma$-algebra of $L^p$ with respect to the natural topology. There is a natural (right) action of $G$ on $L^p$, given by precomposition: $g^* \ell := \ell \circ g$. There is also a natural (left) action, given by conjugation: $c_g \ell := g \circ \ell \circ g^{-1}$.
Does there exist a probability measure on $(L^p,\mathcal B^p)$ which is stationary and ergodic with respect to one (or both) of these actions of $G$?