imagine you have a sequence $\eta_{n}$ of (shift) invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$ that satisfy the following: there are a $0<\delta <1$ and an $N$ such that $$n > N \Rightarrow \sum_{i=1}^{2^{m}}|\eta_{n}(C_{i}^{m}) - \frac{1}{2^{m}}|^{2}<\delta^{2n},$$ for all $m$ where

$C_{1}^{1} = [0], C_{2}^{1} = [1]$,

$C_{1}^{2} =[0,0],...,C_{4}^{2}=[1,1]$,

$C_{i}^{m}$ is the $i^{th}$ cylinder of length $m$, with $i \in \{1,2,...,2^{m}\}$.

So, of course you have much more than $\eta_{n} \rightarrow \lambda$ in the weak* topology, where $\lambda$ is the Bernoulli uniform probability $(\frac{1}{2},\frac{1}{2})$. For instance, you have that all your cylinders are converging (in measure) to $\frac{1}{2^{m}}$ at the same time!

My question is: $h(\eta_{n})\rightarrow \log2$? Are the entropy of the measures growing to $\log2$?

Thanks for your attention

Bruno