International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 10, Pages 585-589
doi:10.1155/S0161171202007494
Translation invariance and finite additivity in a probability
measure on the natural numbers
Department of Mathematics, Box 70663, East Tennessee State University, Johnson City 37614, TN, USA
Received 30 April 2001
Copyright © 2002 Robert Gardner and Robert Price. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Inspired by the “two envelopes exchange paradox,” a finitely
additive probability measure m on the natural numbers is
introduced. The measure is uniform in the sense that
m({i})=m({j}) for all i,j∈ℕ. The measure is
shown to be translation invariant and has such desirable
properties as m({i∈ℕ|i≡0(mod2)})=1/2. For any r∈[0,1], a set A is constructed such that m(A)=r; however, m is not defined on
the power set of ℕ. Finally, a resolution to the two envelopes exchange paradox is presented in terms of m.