A note on representations of Orlicz lattices
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Positivity
A note on representations of Orlicz lattices Pedro Poitevin1 Received: 3 March 2020 / Accepted: 20 October 2020 © Springer Nature Switzerland AG 2020
Abstract In their book Randomly Normed Spaces, Haydon, Levy, and Raynaud proved that every sublattice of a Musielak-Orlicz space L ψ with random sections ψ(·, ω) in a given set D can be represented as a Musielak–Orlicz space L ψ with random sections ψ (·, ω) in the closure (in the product topology) of the convex hull of D. In this note we prove that if L ψ is a Musielak–Orlicz space with random sections ψ (·, ω) in the closure of the convex hull of a set D closed under dilations, then there exists a Musielak–Orlicz space L ψ with random sections ψ(·, ω) in D such that L ψ is a sublattice of L ψ . Furthermore, L ψ can be chosen to have the same density character as L ψ . Keywords Musielak–Orlicz spaces · Orlicz lattices · Continuous logic · Ultraproducts Mathematics Subject Classification 46E30 · 12L10
1 Introduction This note concerns the finite lattice-representability of a class of Musielak–Orlicz spaces, namely the class M O HD of all L ψ (Ω, Σ, μ) with random sections ψ(·, ω) in the closure (in the product topology) of the convex hull of a set D closed under dilations, in another class of Musielak–Orcliz spaces, namely the class of all Musielak– Orlicz spaces L ψ (Ω, Σ, μ) with random sections ψ(·, ω) in D. While it was known that any sublattice of a member of the latter class must be a member of the former (this is something the Representation Theorem for Orlicz Lattices, proved in [4], implies), it was not known whether any member of the former class can be found as a sublattice of a member of the latter. This result is then a kind of converse to the Representation Theorem for Orlicz Lattices.
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Pedro Poitevin [email protected] Mathematics Department, Salem State University, 352 Lafayette St, Salem, MA 01970, USA
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P. Poitevin
2 Preliminaries Decomposable measure spaces A measure space (Ω, Σ, μ) is said to be decomposable if there exists a partition {Ωi : i ∈ I } of Ω into elements Ωi ∈ Σ with 0 < μ(Σi ) < ∞ for all i ∈ I , such that a subset A of Ω belongs to Σ if and only if for all i ∈ I , A ∩ Ωi ∈ Σ, and whenever that happens μ(A) = i∈I μ(A ∩ Ωi ). For the purposes of this note, all measure spaces will be assumed to be decomposable and complete (i.e., every subset of a null set has measure zero). Köthe function spaces If (Ω, Σ, μ) is a (semi-finite) measure space, we denote by L 0 (Ω, Σ, μ) the space of measurable (real or complex) scalar-valued functions defined on Ω, modulo equality μ-almost everywhere. This is a topological Riesz space, for the topology of convergence in measure. A Köthe function space on (Ω, Σ, μ) is a linear subspace X such that: 1. X is an order ideal of L 0 (Ω, Σ, μ); 2. X is equipped with a norm for which it is complete and which is order-compatible (i.e., if f , g ∈ X and | f | ≤ |g|, then f ≤ g); 3. X is order-dense in L 0 (Ω, Σ, μ) (i.e., every A ∈ Σ with μ(A) > 0 contains some B ∈
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