This lecture introduces ergodic theory, focusing on the shift space Ω = Σ^Z, where Σ is a finite set and Z is the group of integers. The shift operator σ shifts the decimal point in an infinite string of elements of Σ. A metric d is defined on Ω, comparing values of strings at each integer, with more weight given to values closer to the decimal point. This metric induces the product topology on Ω, and cylinder sets form a basis for the topology. Ω is a Cantor set, meaning it is compact, totally disconnected, and perfect. The lecture discusses invariant measures under σ, starting with Bernoulli measures, which are defined using probability vectors on Σ. These measures are extended to the full Borel σ-algebra using the Hahn-Kolmogorov extension theorem. Markov measures are another class of invariant measures, defined using a stochastic matrix P and a probability vector p, where p is a left eigenvector of P with eigenvalue 1. The lecture also touches on the construction of elements in Ω with dense orbits, which can be achieved by concatenating finite strings or dense collections of points. The shift operator σ is invertible, and invariant measures are those that are unchanged under σ. The uniqueness of the extension of measures to the Borel σ-algebra is guaranteed by the Hahn-Kolmogorov theorem. The lecture concludes with the definition of Markov measures and their properties, emphasizing their role in ergodic theory.This lecture introduces ergodic theory, focusing on the shift space Ω = Σ^Z, where Σ is a finite set and Z is the group of integers. The shift operator σ shifts the decimal point in an infinite string of elements of Σ. A metric d is defined on Ω, comparing values of strings at each integer, with more weight given to values closer to the decimal point. This metric induces the product topology on Ω, and cylinder sets form a basis for the topology. Ω is a Cantor set, meaning it is compact, totally disconnected, and perfect. The lecture discusses invariant measures under σ, starting with Bernoulli measures, which are defined using probability vectors on Σ. These measures are extended to the full Borel σ-algebra using the Hahn-Kolmogorov extension theorem. Markov measures are another class of invariant measures, defined using a stochastic matrix P and a probability vector p, where p is a left eigenvector of P with eigenvalue 1. The lecture also touches on the construction of elements in Ω with dense orbits, which can be achieved by concatenating finite strings or dense collections of points. The shift operator σ is invertible, and invariant measures are those that are unchanged under σ. The uniqueness of the extension of measures to the Borel σ-algebra is guaranteed by the Hahn-Kolmogorov theorem. The lecture concludes with the definition of Markov measures and their properties, emphasizing their role in ergodic theory.