General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* topology, this, somehow, could help me to prove that $h(\eta_{n})\rightarrow\log p$?
I know it is easy to produce zero-entropy measures that converge in the weak-* topology to $B$. What I would like to know is: having the weak-* topology more some friendly hypothesis on the measures $\eta_{n}$, could I guarantee growth of entropy?