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The study of the Banach-Saks property in Banach spaces has a long and illustrious history. Of late, motivated by applications in financial mathematics, interest has arisen in the Banach-Saks type properties with respect to order convergence. This paper presents a study of order Banach-Saks properties in Banach function spaces, and in particular in rearrangement invariant spaces. Among the results obtained, we provide some sufficient conditions for the (weak) order Banach-Saks property. We also characterize the (weak) order Banach-Saks property in Orlicz spaces. It is also shown that the (weak) order Banach-Saks property is equivalent to its hereditary version., Comment: 19 pages

This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space $\mathcal{X}$. In particular, we show that order closedness, $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$-closedness and $\sigma(\mathcal{X},L^\infty)$-closedness of a law-invariant convex set in $\mathcal{X}$ are equivalent, where $\mathcal{X}_n^\sim$ is the order continuous dual of $\mathcal{X}$. We also provide some application to proper quasiconvex law-invariant functionals with the Fatou property., Comment: 10 pages

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 $X$., Comment: In the 2nd version, we generalized all results to Banach lattices