Measure-theoretic Uniformly Positive Entropy on the Space of Probability Measures

For a homeomorphism $T$ on a compact metric space $X$, a $T$-invariant Borel probability measure $\mu$ on $X$ and a measure-theoretic quasifactor $\tilde{\mu }$ of $\mu$, we study the relationship between the local entropy of the system $(X,\mu ,T)$ and of its induced system $(\mathcal{M}(X),\tilde{\mu },\tilde{T})$, where $\tilde{T}$ is the homeomorphism induced by $T$ on the space $\mathcal{M}(X)$ of all Borel probability measures defined on $X$.