Given a probability measure space (X, Σ , μ) , it is well known that the Riesz space L(μ) of equivalence classes of measurable functions f: X→ R is universally complete and the constant function 1 is a weak order unit. Moreover, the linear functional L∞(μ) → R defined by f↦∫fdμ is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space E with a weak order unit e> 0 which admits a strictly positive order continuous linear functional on the principal ideal generated by e is lattice isomorphic onto L(μ) , for some probability measure space (X, Σ , μ).

A characterization of the vector lattice of measurable functions

Leonetti P
;
2022

Abstract

Given a probability measure space (X, Σ , μ) , it is well known that the Riesz space L(μ) of equivalence classes of measurable functions f: X→ R is universally complete and the constant function 1 is a weak order unit. Moreover, the linear functional L∞(μ) → R defined by f↦∫fdμ is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space E with a weak order unit e> 0 which admits a strictly positive order continuous linear functional on the principal ideal generated by e is lattice isomorphic onto L(μ) , for some probability measure space (X, Σ , μ).
Extended order continuous dual; f-Algebra of L type; Lateral completeness; Measurable functions; Strictly positive order continuous linear functional
Cerreia-Vioglio, S; Leonetti, P; Maccheroni, F
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2142079
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