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233 – 239 10.1007/BF02454172 . [12] Szűcs , Zs. , On the Lebesgue decomposition of positive linear functionals , Proc. Amer. Math. Soc. , to appear

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Abstract  

A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.

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Abstract  

An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL. A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism. These properties were introduced by Schmidt in [5]. In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM. An analogous theorem is derived for Riesz spaces with extension property (E2).

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