On uniform integrability of random variables (Q2567186)

Let \((\Omega, A, P)\) be a probability space, \(F\) be a sub-\(\sigma\)-algebra of \(A.\) \textit{W. Ziȩba} [Probab. Theory Relat. Fields 78, 73--74 (1988; Zbl 0627.60035)] has introduced the concept of uniform \(F\)-integrability for a sequence of random variables \(\{X_n, n \geq 1\}.\) The authors obtain necessary and sufficient conditions for the uniform \(F\)-integrability of the sequence \(\{X_n, n \geq 1\}\) and study properties of such sequences of random variables when they form a conditional amart [\textit{W. Ziȩba}, Theory Probab. Appl. 36, No.~3, 637--639 (1991) and Teor. Veroyatn. Primen. 36, 616--617 (1991; Zbl 0739.60036)].