Die Stonesche Bedingung und die �quivalenz der Integrationstheorien von M. H. STONE und N. BOURBAKI

Vector subspaces and operators with the Stone condition

“…Indeed they proved that any ru-closed vector subspace of E such that u ∧ e ∈ L for all u ∈ L + , is a vector sublattice of E if and only if it is a subalgebra of E. This condition is a type of "Stone condition" introduced by Stone in [6,7] in the context of lattices of real valued functions with pointwise ordering. In [4], Riedrich gives the following definition: a vector sublattice L of a vector lattice E is said to have the Stone condition with respect to a strong order unit e in E whenever u ∈ L implies u ∧ e ∈ L. See also [3] by Pym. In this paper we use the following definitions. Let E be a Φ-algebra in which the unit is denoted by e. We say that a vector subspace L of a E has the Stone condition if u ∈ L implies u ∧ e ∈ L, and that L has the positive Stone…”