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Keywords:
complemented lattices; orthomodular lattices; exhaustive modular functions; measures; extension; Vitali-Hahn-Saks theorem; Nikodým theorems; Liapunoff theorem
complemented lattices; orthomodular lattices; exhaustive modular functions; measures; extension; Vitali-Hahn-Saks theorem; Nikodým theorems; Liapunoff theorem
Summary:
We prove an extension theorem for modular functions on arbitrary lattices and an extension theorem for measures on orthomodular lattices. The first is used to obtain a representation of modular vector-valued functions defined on complemented lattices by measures on Boolean algebras. With the aid of this representation theorem we transfer control measure theorems, Vitali-Hahn-Saks and Nikodým theorems and the Liapunoff theorem about the range of measures to the setting of modular functions on complemented lattices.
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