On special cases of Radon - Nikodym theorem for vector and operator valued measures

TL;DR

This paper provides a clear proof of the Radon-Nikodym theorem for operator-valued measures with separable range, clarifying distinctions between different types of operator measures and addressing gaps in existing literature.

Contribution

It offers a natural, simple proof for the Radon-Nikodym theorem in the context of operator-valued measures with separable range, filling gaps in prior proofs.

Findings

Proof of Radon-Nikodym theorem for operator measures with separable range

Discussion on distinctions between uniform and strong operator measures

Clarification of existing literature gaps and inaccuracies

Abstract

We present a natural and simple proof of the Radon - Nikodym theorem for measures with values in the space of bounded linear operators on a separable Hilbert space. This space is not separable, that is why it is essential to assume in the theorem that the range of the measure is separable. We also discuss distinctions between uniform and strong operator measures (Remark 3.3 to Lemma 3.2, Theorem 3.5 and its Corollary 3.6). Proof of this version of the Radon - Nikodym theorem either is not presented in known literature, or given for special cases and with some inaccuracies (see Comment at the end of the paper), or given in too general form (e. g., Bourbaki) and uses notations that make it hard to apply the theorem.