Ergodic Theory

Abstract

We shall consider the collection of Borel sets, B _i , of a topological space X, which contains the open subsets and is closed under taking complements and countable unions. Such a collection is called a σ-algebra. The Borel sets are always assumed to be measurable. A function μ which associates to each measurable set a non-negative real number so that for each countable collection of disjoint Borel sets B _i one has

$$ \mu ( \cup {B_i}) = \sum\limits_i {\mu ({B_i})} $$

is called a measure. The measure μ is a probability measure if μ(X) = 1 and a σ-finite measure if X can be written as a countable union of sets with finite measure. (Finite measure means that for all B _i , μ(B _i ) < ∞.)