Regular lattice

A conditionally-complete lattice in which the following condition (also called the axiom of regularity) holds: For any sequence $ \{ E _ {n} \} $ of bounded sets for which

$$ \sup E _ {n} \mathop \rightarrow \limits ^ {(} o) a ,\ \ \inf E _ {n} \mathop \rightarrow \limits ^ {(} o) b , $$

there exist finite subsets $ E _ {n} ^ \prime \subset E _ {n} $ with the same property (where $ \rightarrow ^ {(} o) $ denotes convergence in order, cf. also Riesz space). Such lattices are most often met in functional analysis and in measure theory (for example, regular $ K $- spaces and Boolean algebras). They arise naturally in the problem of extending homomorphisms and positive linear operators. In a regular lattice the following two principles hold: a) the diagonal principle (if $ x _ {nm} \rightarrow ^ {(} o) x _ {n} $ and $ x _ {n} \rightarrow ^ {(} o) x $, then $ x _ {nm _ {n} } \rightarrow ^ {(} o) x $ for some sequence of indices $ m _ {n} $); and b) the principle of countability of type (every bounded infinite set contains a countable subset with the same bounds). Conversely, a) and b) together imply the axiom of regularity. Examples of regular lattices are: Any $ K B $- space and, in particular, any $ L _ {p} $, $ 1 \leq p < + \infty $; the Boolean algebra of measurable sets modulo sets of measure 0 in an arbitrary space with a finite countably-additive measure. Other examples of regular Boolean algebras are based on the negation of the Suslin hypothesis.

References

[1]	L.V. Kantorovich, B.Z. Vulikh, A.G. Pinsker, "Functional analysis in semi-ordered spaces" , Moscow-Leningrad (1950) (In Russian)

References

[a1]	W.A.J. Luxemburg, A.C. Zaanen, "Theory of Riesz spaces" , I , North-Holland (1972)