Note on invariant and almost invariant measurable sets (Q2774503)

Let \(\Omega \) be any nonempty set, \(\mathcal A\subset \mathcal P(\Omega)\) be a \(\sigma \)-algebra, \(\mathcal B\) and \(\mathcal C\) be two sub-\(\sigma \)-algebras of \(\mathcal A\) and \(\mathcal P'\) a set of probability measures on \(\mathcal A\). We say that \( \mathcal B \subset \mathcal C [P']\) if for any \(B\in \mathcal B\) there exists \(C\in \mathcal C\) such that \(P(B \bigtriangleup C)=0\) for all \(P\in \mathcal P'\), and we say that \(\mathcal B = \mathcal C [\mathcal P']\) if \(\mathcal B \subset \mathcal C [\mathcal P']\) and \(\mathcal C \subset \mathcal B [\mathcal P']\). NEWLINENEWLINENEWLINEThe main result of this paper is the following: NEWLINENEWLINENEWLINELet \(j=1,2, (\Omega _j, \mathcal A_j, Q_j)\) be probability spaces, \(G_j\) be groups of transformations \(g_j\: \Omega _j\to \Omega _j\) together with a \(\sigma \)-algebra \(\mathcal B (G_j)\) of subsets of \(G_j\) such that the map \((g_j, \omega _j)\to g_j(\omega _j)\) is \((\mathcal B (G_j)\otimes \mathcal A_j, \mathcal A_j)\)-measurable, and \(\mathcal B(G_j)\) admits some \(G_j\)-invariant finitely additive probability. Assume that \(Q\) is \(G_j\)-invariant, and let \(\mathcal B(G_j, \mathcal A_j, Q_j)\subset \mathcal A_j\) be the sub-\(\sigma \)-algebra of all \(Q_j\)-almost \(G_j\)-invariant sets \(A_j\in \mathcal A_j\) and \(\mathcal B (G_1 \times G_2\), \(\mathcal A_1\otimes \mathcal A_2\), \(Q_1 \otimes Q_2)\subset \mathcal A_1\times \mathcal A_2\) be the sub-\(\sigma \)-algebra of all \((Q_1\otimes Q_2)\)-almost invariant sets. Then \(\mathcal B (G_1\times G_2\), \(\mathcal A_1\otimes \mathcal A_2, Q_1 \otimes Q_2)= \mathcal B (G_1, \mathcal A_1, Q_1) \otimes \mathcal B (G_2, \mathcal A_2, Q_2)\) \([\{Q_1\otimes Q_2\}]\). The proof utilizes several technical lemmas, and, in particular, the concept of sufficiency of a sub-\(\sigma \)-algebra \(\mathcal B\) of a \(\sigma \)-algebra \(\mathcal A\) for a set \(\mathcal P'\) of probability measures on \(\mathcal A\). Moreover, an application of the main result to compact groups of transformations is given.