On Some Special Cases of the Radon–Nikodym Theorem for Vector- and Operator-valued Measures

Abstract

This paper presents a proof of the Radon–Nikodym theorem for vector measures with values in a Hilbert space or in the space of bounded linear operators acting from a Hilbert space to a Hilbert space. Assertions for these cases are known ([13], [14], [15]), however they contain some mistakes and inaccuraces (see Concluding Remarks at the end of this paper). Considering operator-valued measures, we emphasize distinctions between the uniform and strong topologies (see Remark 2.3 to Lemma 2.2, Theorem 2.5 and Corollary 2.6). There exist more general versions of the Radon–Nikodym theorem: for measures with values in Banach spaces with boundedly complete Schauder basis or for separable dual Banach spaces (detailed exposition and history can be found, e.g., in [10]). However, we think that a direct and simple proof for the Hilbert space case is of independent interest.

Dedicated to Lev Aronovich Sakhnovich with deep appreciation

Mathematics Subject Classification (2010). 28B05, 46G10.