Elementary Proof of a Theorem of Jean Ville
  Considerable thought has been devoted to an adequate definition of the class of infinite, random binary sequences (the sort of sequence that almost certainly arises from flipping a fair coin indefinitely). The first mathematical exploration of this problem was due to R. Von Mises, and based on his concept of a "selection function." A decisive objection to Von Mises' idea was formulated in a theorem offered by Jean Ville in 1939. It shows that some sequences admitted by Von Mises as "random" in fact manifest a certain kind of systematicity. Ville's proof is challenging, and an alternative approach has appeared only in condensed form. We attempt to provide an expanded version of the latter, alternative argument. 
Elliott H. Lieb, Daniel Osherson and Scott Weinstein