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

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