Generic properties of invariant measures of full-shift systems over perfect Polish metric spaces

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, a product space whose alphabet is perfect and separable metric (thus, complete uncountable). More specifically, show that set upper Hausdorff dimension equal to zero lower packing infinity dense $G_\delta$ subset $\mathcal{M}(T)$, $T$-invariant endowed weak topology. We also rate recurrence $\mathcal{M}(T)$. Furthermore, quantitative waiting time indicator residual