How to extend Carathéodory's theorem to lattice-valued functionals

Abstract

Substituting in the definition of outer measure the addition with the maximum (or the supremum, or the join) operation we obtain a new set function called retuo measure. It is proved that every retuo measure is an outer measure. We give necessary and sufficient conditions for a set function to be a retuo measure. Similarly as in the case of outer measure, we propose a way to construct retuo measures. We consider some theoretical applications for constructed pairs of outer and retuo measures in the image of the Hausdorff measure and dimension.

Keywords

• outer mearsure
• sigma-algebra
• measurable sets
• Hausdorff measure
• Hausdorff dimension
• Caratheodory-type theorem

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