Ergodic Theory

  • Igor Nikolaev
Chapter
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 41)

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 ) < ∞.)

Keywords

Entropy Manifold 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Igor Nikolaev
    • 1
  1. 1.The Fields Institute for Research in Mathematical SciencesTorontoCanada

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