Boundedness and convergence theorems on non-Boolean structures
Anna Avallone, Paolo Vitolo · 2007
In the last years many authors investigated the validity of Brooks‐Jewett, Vitali‐Hahn‐Saks and Nikodym Boundedness theorems for measures on more general structures than Boolean rings. In particular, De Lucia and Pap in 1995 proved Brooks‐Jewett theorem for measures on eect algebras with the Sequential Completeness Property (SCP) and gives a characterization of the uniform boundedness of a set of functions on such structures. It follows from a result of D’Andrea, De Lucia and Morales that Nikodym Boundedness theorem and Vitali‐Hahn‐Saks theorems fail for measures on eect algebras, but Avallone proved that these theorems hold for modular measures on lattice-ordered eect algebras (otherwise called D-lattices) with the SCP. We prove that Brooks‐Jewett, Vitali‐Hahn‐Saks and Nikodym Boundedness theorems also hold for modular measures on lattice-ordered eect