No-Free-Lunch theorems in the continuum

No-Free-Lunch Theorems state, roughly speaking, that the performance of all search algorithms is the same when averaged over all possible objective functions. This fact was precisely formulated for the first time in a now famous paper by Wolpert and Macready, and then subsequently refined and extended by several authors, usually in the context of a set of functions with finite domain and codomain. Recently, Auger and Teytaud have studied the situation for continuum domains. In this paper we provide another approach, which is simpler, requires less assumptions, relates the discrete and continuum cases, and we believe that clarifies the role of the cardinality and structure of the domain.