Measure Theory and Ergodic Theory
Backlinks
The foundational knowledge relating to Measure & Ergodic Theory, that has not been covered elsewhere, was provided by Robertson (2023).
Definition 1 (Gleason (2010)) A topological measure space \((X,\SigmaAlgebra{B},\Measure)\) is a topological space \((X,\Topology)\) such that \(\SigmaAlgebra{B}\) is generated by the open sets defined by the topology \(\Topology\), i.e., \(\SigmaAlgebra{B}=\text{Borel}(X)=\SigmaAlgebraGenerator\), and \(\Measure\) is a measure on this space.
A Borel measure is the measure \(\Measure\) on a topological measure space \((X,\SigmaAlgebra{B},\Measure)\) where \((X,\Topology)\) is Hausdorff.
A regular (Borel) measure is a measure on a Borel measure space \((X,\SigmaAlgebra{B},\Measure)\) such that the following hold:A left-Haar measure [or right-Haar measure] on a topological group \((\Group,\Topology)\) is a non-zero regular Borel measure \(\Measure\) on \(\Group\) such that \(\Measure(\GroupOperation{\GroupElement}{B})=\Measure(B)\) [or \(\Measure(\GroupOperation{B}{\GroupElement})=\Measure(B)\)] for all \(\GroupElement\in\Group\) and \(B\in\sigma(\Topology)\).