Translation Invariance and Finite Additivity in a Probability Measure on the Natural Numbers

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.