Measures on effect algebras

We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on which σ-additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.

RETRACTED ARTICLE: An augmented Riesz decomposition method for sharp estimates of certain boundary value problem

In this paper, by using an augmented Riesz decomposition method, we obtain sharp estimates of harmonic functions with certain boundary integral condition, which provide explicit lower bounds of functions harmonic in a cone. The results given here can be used as tools in the study of integral equations.