International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 10, Pages 585-589

Translation invariance and finite additivity in a probability measure on the natural numbers

Robert Gardner and Robert Price

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.


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. The measure is shown to be translation invariant and has such desirable properties as m({i|i0(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 . Finally, a resolution to the two envelopes exchange paradox is presented in terms of m.