Measure Theory and Ergodic Theory
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The foundational knowledge relating to Measure & Ergodic Theory, that has not been covered elsewhere, was provided by Robertson (2023).
Definition 1 (Gleason, 2010, sect. 2) 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 \(B\in\sigma(\Topology)\).
1 Eigenfunctions
Let \((X,\Borel{X},\Measure,T)\) be a measure-preserving system.
Definition 2 (Kra et al., 2026, sect. 2) A nonzero function \(f\in\text{L}^2(X,\Borel{X},\Measure)\) is a measurable eigenfunction with eigenvalue \(\theta\in\mathbb{C}\) if \(f(Tx)=\theta f(x)\) holds for \(\Measure\)-almost every \(x\in X\).
Definition 3 (Kra et al., 2026, sect. 2) A nonzero function \(f\in\ContinuousFunctionSpace{X}\) is a topological eigenfunction with eigenvalue \(\theta\in\mathbb{C}\) if \(f(Tx)=\theta f(x)\) for all \(x\in X\).
2 Discrete Spectrum
Proposition 1 (Jamneshan and Kreidler, 2025, Corollary 5.1.11) Let \(U:\Group\rightarrow\mathscr{U}(\HilbertSpace)\) be a unitary representation of an abelian group \(\Group\). Then \[ \HilbertSpace_\text{ds}=\bar{\text{lin}}\bigcup_{\chi\in\Dual{\Group}}\text{ker}(\chi-U). \]
3 Mixing
Definition 4 (Kra et al., 2026, sect. 2) We define \(\EigenAlgebra\) to be the sub-\(\sigma\)-algebra of \(\Borel{X}\) where \(\text{L}^2(X,\EigenAlgebra,\Measure)\) is the closed subspace of \(\text{L}^2(X,\Borel{X},\Measure)\) spanned by its measurable eigenfunctions.
Proposition 2 (cf. Kra et al., 2026, sect. 2) For any \(E\in\Borel{X}\) and \(\Measure\)-almost every \(x\in E\), we have \(\mathbb{E}(\boldsymbol{1}_E\mid\EigenAlgebra)(x)>0\).
Theorem 1 (Kra et al., 2026, Theorem 2.4) For any \(f\in\text{L}^2(X,\Borel{X},\Measure)\), we have \[ f\text{is weak mixing}\quad\iff\quad\mathbb{E}(f\mid\EigenAlgebra)=0. \]