I saw an example to show how to calculate the Measure-theoretic entropy, and here is the example:
Let $T:X\to X$ be the doubling map $T(x)=2x\;(mod\;1)$ and there is a partition $\alpha=\{[0,\frac{1}{2}),[\frac{1}{2},1)\}$
$\lor_{i=0}^{n-1}T^{-i}\alpha=\{[\frac{i}{2^n},\frac{i+1}{2^n})\,:i=0,...2^n-1\}$ and the example just say $\alpha$ is the generator, then calculate that the result is log2.
What I don't unstand is that why $\alpha$ is a generator for T ?
According to Kolmogorov-Sinai Theorem, is this $\lor_{i=0}^{\infty}T^{-i}\alpha$ generating $\sigma$ algebra of [0,1)? I think it's not, so I don't unstand what kind of $\sigma$ algebra it generates and why $\alpha$ can be a generator.
I suspect the problem is my thought about X, maybe X is not [0,1)?
$$\lor_{i=0}^{n-1}T^i\alpha=\{[\frac{i}{2^n},\frac{i+1}{2^n})\,:i=0,...2^n-1\}$$ and the union of these over $n$ generates the Borel $\sigma$ algebra.