Independence of events and of random variables

The phenomenon of independence of random variables is shown to be singular in that, e.g., there are both finite and infinite sample spaces on which two random variables can be independent iff one is constant. Furthermore on 0,1], with Lebesgue measure for probability, the usual function spaces contain dense subsets each member of which is independent only of constants. Finally, the requirement of independence among a set of orthonormal functions is shown to imply, in all but trivial instances, that the orthogonal complement of the space is infinite-dimensional.