# Basic Ergodic Theory

Definition 1. (Measure-preserving dynamical system) A measure-preserving dynamical system $$(X, \mathcal{X}, \mu, T)$$ is defined by specifying a probability space $$(X, \mathcal{X}, \mu)$$ and a measure-preserving transformation (isomorphism) $$T$$ on it:

• $$X$$ is a set.
• $$\mathcal{X}$$ is a σ-algebra over $$X$$.
• $$\mu : \mathcal{X} \to [0,1]$$ is a probability measure.
• $$T : X \to X$$ is a measurable transformation which preserves the measure $$\mu$$, i.e., $$\forall E \in \mathcal{X}, \mu(T(E)) = \mu(E)$$.

Definition 2. (Measure-theoretic entropy; Kolmogorov-Sinai entropy) The entropy of a partition $$Q=\{Q_1,\dots,Q_k\}$$ is defined as $\operatorname{H}(Q) = -\sum_{m=1}^k \mu(Q_m) \log \mu(Q_m)$ The measure-theoretic entropy of a dynamical system $$(X,\mathcal{X},T,\mu)$$ with respect to a partition $$Q=\{Q_1,\dots,Q_k\}$$ is defined as $h_\mu(T,Q) = \lim_{N \to \infty} \frac{1}{N} \operatorname{H}\left(\bigvee_{n=0}^N T^{-n}Q\right)$ The measure-theoretic entropy of a dynamical system $$(X,\mathcal{X},T,\mu)$$ is defined as $h_\mu(T) = \sup_Q h_\mu(T,Q)$ where the supremum is taken over all finite measurable partitions $$Q$$.