The present work continues investigation of the capacities of measurement (quantum-classical) channels in the most general setting, initiated in~cite{HCT}. The proof of coding theorems is given for the classical capacity and entanglement-assisted cla
ssical capacity of the measurement channel with arbitrary output alphabet, without assuming that the channel is given by a bounded operator-valued density.
In this paper a general definition of quantum conditional entropy for infinite-dimensional systems is given based on recent work of Holevo and Shirokov arXiv:1004.2495 devoted to quantum mutual and coherent informations in the infinite-dimensional ca
se. The properties of the conditional entropy such as monotonicity, concavity and subadditivity are also generalized to the infinite-dimensional case.
