On the Riesz Integral Representation of Additive Set-Valued Maps (II)

Abstract

<jats:p> Let T be a compact topological space, and let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_+(T)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be the space of all non-negative continuous real-valued functions defined on T endowed with the topology of uniform convergence. We prove the Riesz integral representation for continuous additive and positive set-valued maps defined on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_+(T)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> with values in the space cc(E) of all weakly compact convex non-empty subsets of a Banach space E. As an application we give a generalization of Dunford-Schwartz's result on the Riesz integral representation for any continuous set-valued map (not necessary positive). </jats:p>