  Coherence is a cornerstone of quantum theory and a prerequisite for the
advantage of quantum technologies. In recent work, the notion of coherence with
respect to a general quantum measurement (POVM) was introduced and embedded
into a resource-theoretic framework that generalizes the standard resource
theory of coherence. In particular, POVM-incoherent (free) states and
operations were established. In this work, we explore features of this
framework which arise due to the rich structure of POVMs compared to projective
measurements. Moreover, we introduce a rigorous, probabilisitic framework for
POVM-based coherence measures and free operations. This leads to the
introduction of new, strongly monotonic resource measures that neatly
generalize well-known standard coherence measures. Finally, we show that the
relative entropy of POVM-coherence is equal to the cryptographic randomness
gain, providing an important operational meaning to the concept of coherence
with respect to a general measurement.
