On the representation of tight functionals as integrals

On the representation of tight functionals as integrals We consider a vector lattice $$\mathcal L$$ of bounded real continuous functions on a topological space $$X$$ that separates the points of $$X$$ and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on $$\mathcal L$$ can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On the representation of tight functionals as integrals

, Volume 17 (4) – Feb 3, 2013
13 pages
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Publisher
Springer Basel
Copyright
Copyright © 2013 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0223-y
Publisher site
See Article on Publisher Site

Abstract

We consider a vector lattice $$\mathcal L$$ of bounded real continuous functions on a topological space $$X$$ that separates the points of $$X$$ and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on $$\mathcal L$$ can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures.

Journal

PositivitySpringer Journals

Published: Feb 3, 2013

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