Continuity condition for concave functions on convex $\mu$-compact sets and its applications in quantum physics

A method of proving local continuity of concave functions on convex set possessing the $\mu$-compactness property is presented. This method is based on a special approximation of these functions. The class of $\mu$-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which particular results well known for compact sets can be generalized. Applications of the obtained continuity conditions to analysis of different entropic characteristics of quantum systems and channels are considered.