In this short note, we show that every convex, order bounded above functional on a Banach lattice is automatically norm continuous. This improves a result in {RS06} and applies to many deviation and variability measures. We also show that an order-continuous, law-invariant functional on an Orlicz space is strongly consistent everywhere, extending a result in {KSZ14}.
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Gao et al. (2024) studied this question.
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