Radon-Nikodým Derivatives of Hilbert Space Valued Measures

Abstract

The weak Radon-Nikodým derivative of a Hilbert space valued measure is introduced. We obtain a necessary and sufficient condition for the existence of such a measure and study some of its properties. Integration of a scalar valued function with respect to a Hilbert space valued measure having a weak Radon-Nikodým derivative is seen to be related to the usual Dunford-Schwartz integral. Finally, we examine the case where the measure has values in the class of Hilbert-Schmidt operators.

Key Words:

• Hilbert space valued measures
• Weak Radon-Nikodým derivatives
• v-boundedness