In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{ba}(\Sigma ,(X,Y))$. This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{ba}(\Sigma ,₁(X,Y))$. This result has interesting applications in optimization and control theory as illustrated by several examples.

We generalize the theory of Wiener amalgam spaces on locally compact groups to quasi-Banach spaces. As a main result we provide convolution relations for such spaces. Also we weaken the technical assumption that the global component is invariant under right translations, which is new even for the classical Banach space case. To illustrate our theory we discuss in detail an example on the ax+b group.