Factor Maps
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Definition 1 (Kra et al., 2022, Definition 2.1) For a system \((X,\Measure,T)\) we say that the system \((Y,\nu,S)\) is a measurable factor of \((X,\Measure,T)\) if there is a measurable map \(\ProjectionMap:X\rightarrow Y\), the measurable factor map, such that \(\ProjectionMap(\Measure)=\nu\) and \((S\circ\ProjectionMap)(x)=(\ProjectionMap\circ T)(x)\) for \(\Measure\)-almost every \(x\in X\).
Definition 2 (Jamneshan and Kreidler, 2025, Lemma 8.11) Let \((X\SigmaAlgebra{B}),(Y,\SigmaAlgebra{C})\) be probability spaces and let \(U:\text{L}^2(X)\rightarrow\text{L}^2(Y)\) be a Markov embedding. The conditional expectation of \(f\in\text{L}^2(X)\) on \((Y)\) is the orthogonal projection of \(f\) on \(U(\text{L}^2(Y))\), denoted \(\Expectation(f\mid Y)\), such that
- (Linearity) \(f\mapsto\Expectation(f\mid Y)\) is a linear operator from \(\text{L}^2(X)\) to \(\text{L}^2(Y)\), i.e., \[\Expectation(f+g\mid\SigmaAlgebra{C})=\Expectation(f\mid\SigmaAlgebra{C})+\Expectation(g\mid\SigmaAlgebra{C}).\]
- (Non-Negative) If \(f\geq0\), then \(\Expectation(f\mid Y)\geq0\).
- If \(f\in \text{L}^2(Y)\), then \(\Expectation(U(f)\mid Y)=f\). In particular, \(\Expectation(\mathbb{1}\mid Y)=\mathbb{1}\).
- If \(f\in\text{L}^2(X)\) and \(g\in\text{L}^\infty(Y)\), then \(\Expectation(fU(g)\mid Y)=g\Expectation(f\mid Y)\).
- For \(f\in\text{L}^2(X)\), \[\int_X f\ \mathrm{d}\Measure_X=\int_Y\Expectation(f\mid Y)\ \mathrm{d}\Measure_Y. \]
- For \(f\in\text{L}^2(X)\), the condtional expectation \(\Expectation(f\mid Y)\) is the unique element of \(\text{L}^2(Y)\) satisfying \[\int_Y\Expectation(f\mid Y)h\ \mathrm{d}\Measure_Y=\int_Xf(U(h))\ \mathrm{d}\Measure_X, \]for all \(h\in\text{L}^\infty(Y)\).
- For \(f\in\text{L}^2(X)\), \[|\Expectation(f\mid Y)|^2\leq\Expectation(|f|^2\mid Y). \]
- The conditional expectation operator extends to a linear operator from \(\text{L}^1(X)\) to \(\text{L}^1(Y)\) satisfying the properties (1)-(5). Moreover, it maps each \(\text{L}^p(X)\) to \(\text{L}^p(Y)\), for \(1\leq p\leq\infty\), with \(||\Expectation(f\mid Y)||)_{\text{L}^p(Y)}\leq||f||_{\text{L}^p(X)}\) for every \(f\in\text{L}^p(X)\).
- (Monotone Convergence) If \(0\leq f_1\leq f_2\leq...\) is a monotone sequence and \(f\) is an element in \(\text{L}^2(X)\) such that \((f_n)_{n\in\mathbb{N}}\) converges to \(f\) almost everywhere, then \(\Expectation(f_n\mid Y)\) converges to \(\Expectation(f\mid Y)\) almost everywhere.
- (Dominated Convergence) If \((f_n)_{n\in\mathbb{N}}\) is a sequence and \(f\) is an element in \(\text{L}^2(X)\) such that \(f_n\) converges to \(f\) almost everywhere and there is \(g\in\text{L}^1(X)\) with \(|f_n|\leq g\) almost everywhere for all \(n\), then \(\Expectation(f_n\mid Y)\) converges to \(\Expectation(f\mid Y)\) almost surely.
In the case that \(Y=X\), then we instead denote the conditional expectation as \(\Expectation(f\mid\SigmaAlgebra{C})\).
Whilst looking at conditional expectations and understanding property (5), we remarked that \(\Expectation(f\mid\SigmaAlgebra{C})\) averages a function \(f:X\rightarrow\SigmaAlgebra{C}\) in \((X,\SigmaAlgebra{B})\) on each subset in \(\SigmaAlgebra{C}\).
Definition 3 (Kra et al., 2022) Fix a system \((X,\Measure,\Group)\) and a probability space \((\Omega,\nu)\). Whenever we have a map \(\omega\mapsto\Measure_\omega\) from \(\Omega\) to \(\mathcal{M}(X)\), the set of probability measures on \(X\), with the properties
- \(\omega\mapsto\Measure_\omega(B)\) is measurable for every Borel set \(B\subset X\),
- \(\int_Xf\ \mathrm{d}\Measure=\int_\Omega\int_Xf\ \mathrm{d}\Measure_\omega\ \mathrm{d}\nu(\omega)\) for all measurable and bounded \(f:X\rightarrow\mathbb{C}\),
we call \(\omega\mapsto\Measure_\omega\) a disintegration of \(\Measure\).
Theorem 1 If \(\pi:(X,\SigmaAlgebra{B},\Measure)\rightarrow(Y,\SigmaAlgebra{C},\nu)\) is a measurable factor map, then there exists a disintegration \(y\mapsto\Measure_y\) of \(\Measure\) defined on \((Y,\nu)\) such that \[\Expectation(f\mid Y)(y)=\int_Xf\ d\Measure_y \] for \(\nu\)-almost every \(y\in Y\), whenever \(f:X\rightarrow\mathbb{C}\) is measurable and bounded. Furthermore, if \(y\mapsto\eta_y\) is another such disintegration, then \(\eta_y=\Measure_y\) for \(\nu\)-almost every \(y\in Y\).
Definition 4 (Kra et al., 2022, Definition 2.7) Let \((X,\Measure,\Group)\) be a system and \(\SigmaAlgebra{C}\) be a \(\sigma\)-algebra of \(G\)-invariant Borel sets. An of \(\Measure\) is a disintegration \(x\mapsto\Measure_x\) of \(\Measure\) defined on \((X,\Measure)\) such that \[\int_Xf\ d\Measure_x=E(f\mid\SigmaAlgebra{C})(x) \] holds for \(\Measure\)-almost every \(x\in X\), whenever \(f:X\rightarrow\mathbb{C}\) is measurable and bounded.
Notice that, by Theorem 1, we immediately have the property that, if \(x\mapsto\nu_x\) is another ergodic decomposition of \((X,\Measure,\Group)\), \(\Measure_x=\nu_x\) for \(\Measure\)-almost every \(x\in X\).
In working on Exercise 8.5 provided by Jamneshan and Kreidler (2025), we also reach the following result:
Theorem 2 (cf. Jamneshan and Kreidler, 2025, Exercise 8.5) For a system \((X,\Measure,\Group)\), there exists an ergodic decomposition \(x\mapsto\Measure_x\) where, for \(\Measure\)-almost every \(x\in X\), the measure \(\Measure_x\) is \(\Group\)-invariant and ergodic.
This is proved by starting with Definition 3 and Theorem 1 and using the assumptions that \(Y=X\), \(\nu=\Measure\), and \(\SigmaAlgebra{C}\) is the \(\sigma\)-algebra of \(\Group\)-invariant Borel sets. We then use the fact that \(\text{L}^2(Y,\SigmaAlgebra{C})\subseteq\text{L}^2(X,\SigmaAlgebra{B})\) to conclude that there exists a Markov embedding \[U:\text{L}^2(Y,\Sigma_Y,\Measure_Y)\rightarrow\text{L}^2(X,\text{Bor}(X),\Measure_X)\] where \(U(\text{L}^2(Y,\Sigma_Y,\nu))=\text{L}^2(X,\Sigma_Y,\Measure)\). We then show that \((X,\SigmaAlgebra{B},\Measure_y,\Group)\) is a measure-preserving system by showing that \((U_\gamma)_*\Measure_y=\Measure_y\) for every \(\gamma\in\Group\) and almost every \(y\in Y\) using Theorem 1. We conclude the proof by proving ergodicity by arriving to the conclusion that \(\Measure_y(E)=\mathbb{1}_E(y)\in\{0,1\}\) for some \(E\in\SigmaAlgebra{C}\).