Let X be a Banach lattice. A well-known problem arising from the theory of risk measures asks when order closedness of a convex set in X implies closedness with respect to the topology \(\sigma (X,X_n^\sim )\), where \(X_n^\sim \) is the order continuous dual of X. Motivated by the solution in the Orlicz space case, we introduce two relevant properties: the disjoint order continuity property (DOCP) and the order subsequence splitting property (OSSP). We show that when X is monotonically complete with OSSP and \(X_n^\sim \) contains a strictly positive element, every order closed convex set in X is \(\sigma (X,X_n^\sim )\)-closed if and only if X has DOCP and either X or \(X_n^\sim \) is order continuous. This in turn occurs if and only if either X or the norm dual \(X^*\) of X is order continuous. We also give a modular condition under which a Banach lattice has OSSP. In addition, we also give a characterization of X for which order closedness of a convex set in X is equivalent to closedness with respect to the topology \(\sigma (X,X_{uo}^\sim )\), where \(X_{uo}^\sim \) is the unbounded order continuous dual of X.

中文翻译：

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Banach格上凸集的闭性

令X为Banach晶格。由风险测度理论引起的一个众所周知的问题是，X中凸集的顺序封闭性何时暗示相对于拓扑\（\ sigma（X，X_n ^ \ sim）\）的封闭性，其中\（X_n ^ \ sim \）是X的顺序连续对偶。受Orlicz空间中的解的启发，我们引入了两个相关的属性：不相交的顺序连续性（DOCP）和顺序子序列分裂属性（OSSP）。我们证明当X与OSSP和\（X_n ^ \ sim \）单调完成时包含一个严格的正元素，X中的每个阶闭合凸集都是\（\ sigma（X，X_n ^ \ sim）\）-仅当X具有DOCP并且X或\（X_n ^ \ sim \）为订单连续。发生这反过来如果且仅当X或常态双\（X ^ * \）的X是顺序连续的。我们还给出了Banach晶格具有OSSP的模块化条件。此外，我们还给出的表征X为其中在凸集的顺序闭性X相当于封闭性相对于所述拓扑\（\ sigma（X，X_ {uo} ^ \ sim）\），其中\（X_ {uo} ^ \ sim \）是X的无穷阶连续对偶。