Simplifying conditional expectations (Q1808644)

Let \({\mathcal G}\) be a \(\sigma\)-field, and assume \(\{{\mathcal F}_a:a \in A\}\) to be a collection of \(\sigma\)-fields conditionally independent given \({\mathcal G}\). Fix \(a_0\in A\), put \({\mathcal F}^{(a_0)}= \bigvee_{a\in A-\{a_0\}} {\mathcal F}_a\), and let \({\mathcal D}\) be a \(\sigma\)-field with \({\mathcal D}\subset {\mathcal F}_{a_0} \vee{\mathcal G}\). The authors prove that \(E[I(F)\mid {\mathcal D}\vee{\mathcal F}^{(a_0)} \vee{\mathcal G}]= E[I(F)\mid {\mathcal D}\vee {\mathcal G}]\) a.s., for any \(F\in {\mathcal F}_{a_0} \vee{\mathcal G}\). If \({\mathcal D}\) is the trivial \(\sigma\)-field, the result coincides with the ``only if'' part of Theorem 9.2.1 of \textit{K. L. Chung} [``A course in probability theory'' (1974; Zbl 0345.60003)].