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

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.