Ordered representations of spaces of integrable functions

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 13; no. 1; pp. 129 - 143
• Main Authors: Fernández, Antonio, Mayoral, Fernando, Naranjo, Francisco, Sánchez–Pérez, Enrique A.
• Format: Journal Article
• Language: English
• Published: Basel Birkhäuser-Verlag 01.02.2009; Springer; Springer Nature B.V
• Subjects: Algebra; Banach spaces; Calculus of Variations and Optimal Control; Optimization; Combinatorics. Ordered structures; Econometrics; Exact sciences and technology; Fourier Analysis; Functional analysis; Mathematical analysis; Mathematical functions; Mathematics; Mathematics and Statistics; Operator Theory; Order, lattices, ordered algebraic structures; Potential Theory; Sciences and techniques of general use; Studies; Vector space; Vector measure; Representation; Secondary 47B65, 47H07; Banach function space; Primary 46E30, 46G10; Function space; Mathematical analysis; Ordered set; Measurable space; Banach space; Ordered space; N order
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-008-2211-1

Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L 1 ( m ). In this paper we analyze the set of (countably additive) vector measures n satisfying that L 1 ( n ) =  L 1 ( m ). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L 1 ( m ) associated to downward directed subsets of the set of all the representations.