Acta Mathematica Hungarica

, Volume 122, Issue 1–2, pp 161–172 | Cite as

On characterizations of sup-preserving functionals

Article
First Online:
Received:
Revised:
Accepted:
  • 28 Downloads

Abstract

Let (E, ≦) be a vector lattice and E + be the set of all nonnegative elements of E. We investigate M-functionals from E + into ℝ+, that is functions A: E + → ℝ+ such that
$$ \Lambda (f \vee g) = \Lambda (f) \vee \Lambda (g),\Lambda (\alpha f) = \alpha \Lambda (f) $$
for α ≧ 0 and f, g ɛ E +.
Let X be a set and Σ be an algebra of subsets of X. By an M-measure we understand the function μ: Σ → ℝ+ such that μ(\( \not 0 \)) = 0 and
$$ \mu (A \cup B) = \mu (A) \vee \mu (B)forA,B \in \Sigma ). $$
The main result of the paper is a Riesz type theorem. We prove that every M-functional on C(X, ℝ)+ can be expressed in terms of M-measure.

Key words and phrases

Banach lattice M-space M-pseudonorm Riesz theorem 

2000 Mathematics Subject Classification

46G12 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    W. Rudin, Functional Analysis, McGraw-Hill (New York, 1973).MATHGoogle Scholar
  2. [2]
    H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag (Berlin, 1974).MATHGoogle Scholar
  3. [3]
    Z. Semadeni, Banach Spaces of Continuous Functions, Volume I, PWN — Polish Scientific Publishers (Warszawa, 1971).Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of RzeszówRzeszówPoland
  2. 2.Institute of MathematicsJagiellonian UniversityKrakówPoland

Personalised recommendations