Egoroff, σ, and convergence properties in some archimedean vector lattices

• Autorzy: A. W. Hager , J. van Mill
• Języki publikacji: EN

Abstrakty

EN

An archimedean vector lattice A might have the following properties:
(1) the sigma property (σ): For each ${aₙ}_{n∈ ℕ} conA⁺$ there are ${λₙ}_{n ∈ ℕ} ⊆ (0,∞)$ and a ∈ A with λₙaₙ ≤ a for each n;
(2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u.
The conjunction of these two is called strongly Egoroff.
We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent to this property of X: (E) the intersection of any sequence of dense cozero-sets contains another. (In case X is zero-dimensional, (E) holds iff the clopen algebra clopX of X is a 'Egoroff Boolean algebra'.)
A crucial part of the proof is this theorem about any compact X: For any countable intersection of dense cozero-sets U, there is uₙ ↓ 0 in C(X) with {x ∈ X: uₙ(x) ↓ 0} = U. Then, we make a construction of many new X with (E) (thus, dually, strongly Egoroff D(X)), which can be F-spaces, connected, or zero-dimensional, depending on the input to the construction. This results in many new Egoroff Boolean algebras which are also weakly countably complete.

Kategorie tematyczne

• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces
• 06E15: Stone spaces (Boolean spaces) and related structures
• 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
• 46A40: Ordered topological linear spaces, vector lattices
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 54G05: Extremally disconnected spaces, F -spaces, etc.
• 54G10: P -spaces
• 28A05: Classes of sets (Borel fields, σ -rings, etc.), measurable sets, Suslin sets, analytic sets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2015
• Tom: 231
• Numer: 3
• Strony: 269-285

Daty

wydano

2015

Twórcy

autor

A. W. Hager

• Department of Mathematics, Wesleyan University, Middletown, CT 06459, U.S.A.

autor

J. van Mill

• KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands

Identyfikatory

DOI

10.4064/sm8363-2-2016