On the Properties of a Collective
Z. W. Birnbaum, Herbert S. Zuckerman · American Journal of Mathematics · 1940
Postulate 2. If an infinite subsequence of the total sequence is formed by a selection then, for this subsequence, the same limits exist and their values remain unchanged, lim n'0/n = w0, lim n'/'n = w1. n-*oo The numbers w0 and w1 are called probabilities of tlhe appearance of the labels O and 1 in the collective. These postulates have become the object of considerable discussion. Most of these discussionis have centred aroulld the secoind postulate and a number of investigations have been made in attempts to prove the consistency of the concept of a collective, in connection with the difficulties encountered in interpreting this postulate.3 It is the aim of the present paper to prove that a sequence which fulfills the first postulate, fulfills also, generally speaking, the second postulate. The precise formulation of this statement is given in the following