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