Alonso, Alberto; Brambila-Paz, Fernando \(L^p\)-continuity of conditional expectations. (English) Zbl 0917.60034 J. Math. Anal. Appl. 221, No. 1, 161-176 (1998). A necessary and sufficient condition on a sequence \(({\mathcal A}_n)_{n\in\mathbb{N}}\) of \(\sigma\)-subalgebras is given such that the conditional expectations \(E(f\mid {\mathcal A}_n)\) converge in \(L^p\)-norm \((1\leq p<\infty)\). This result generalizes the convergence theorem of the \(L^p\)-martingales, the Fetter and Boylan equi-convergence theorems. Some examples and counterexamples are also given. Reviewer: F.Weisz (Budapest) Cited in 1 ReviewCited in 8 Documents MSC: 60F25 \(L^p\)-limit theorems 60G48 Generalizations of martingales Keywords:convergence of conditional expectations; martingales PDFBibTeX XMLCite \textit{A. Alonso} and \textit{F. Brambila-Paz}, J. Math. Anal. Appl. 221, No. 1, 161--176 (1998; Zbl 0917.60034) Full Text: DOI References: [1] Alonso, A., A counterexample on the continuity of conditional expectations, J. Math. Anal. Appl., 129, 1-5 (1988) · Zbl 0641.60053 [2] Boylan, E., Equiconvergence of martingales, Ann. Math. Statist., 42, 552-559 (1971) · Zbl 0218.60049 [3] F. Brambila, A. Alonso, Continuity of conditional expectations; F. Brambila, A. Alonso, Continuity of conditional expectations · Zbl 0917.60034 [4] Fetter, H., On the continuity of conditional expectations, J. Math. Anal. Appl., 61, 227-231 (1977) · Zbl 0415.60003 [5] Neveu, J., Mathematical Foundations of the Calculus of Probability (1965), Holden-Day: Holden-Day San Francisco · Zbl 0137.11301 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.