Compact spaces that result from adequate families of sets

We consider compact spaces defined by adequate families of sets as well as continuous images of such spaces which are called AD-compact. The class of AD-compact spaces contains all polyadic spaces. We note some general properties of AD-compact spaces. We prove that there are nonpolyadic AD-compact spaces having a strictly positive measure. We also show that some results on Banach spaces C(K) valid for a dyadic K may be extended to K being AD-compact.