This section introduces the basics of ergodic theory, focusing on the shift operator and its dynamics on the space of infinite sequences. The set Ω is defined as Σ^Z, where Σ is a finite set, and the shift operator σ shifts the sequence one position to the right. The metric d on Ω is introduced, which compares the values of two sequences at each integer, giving more weight to values closer to the decimal point. This metric induces the product topology on Ω. The section also discusses the construction of cylinder sets, which are clopen subsets of Ω determined by a specific integer and a finite word. It mentions that Ω is homeomorphic to a Cantor set, and provides two methods for constructing sequences with dense orbits under the shift operator. Finally, the section covers invariant measures on Ω, including Bernoulli measures and Markov measures. Bernoulli measures are defined using probability vectors and the Hahn-Kolmogorov extension theorem to extend them to the Borel σ-algebra. Markov measures are defined using a stochastic matrix and a probability vector, and the uniqueness of the leading eigenvalue of the matrix ensures the existence of a unique left eigenvector. An exercise is provided to prove that these measures are shift-invariant.This section introduces the basics of ergodic theory, focusing on the shift operator and its dynamics on the space of infinite sequences. The set Ω is defined as Σ^Z, where Σ is a finite set, and the shift operator σ shifts the sequence one position to the right. The metric d on Ω is introduced, which compares the values of two sequences at each integer, giving more weight to values closer to the decimal point. This metric induces the product topology on Ω. The section also discusses the construction of cylinder sets, which are clopen subsets of Ω determined by a specific integer and a finite word. It mentions that Ω is homeomorphic to a Cantor set, and provides two methods for constructing sequences with dense orbits under the shift operator. Finally, the section covers invariant measures on Ω, including Bernoulli measures and Markov measures. Bernoulli measures are defined using probability vectors and the Hahn-Kolmogorov extension theorem to extend them to the Borel σ-algebra. Markov measures are defined using a stochastic matrix and a probability vector, and the uniqueness of the leading eigenvalue of the matrix ensures the existence of a unique left eigenvector. An exercise is provided to prove that these measures are shift-invariant.