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

International Journal of Mathematics and Mathematical Sciences, Mar 2019

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.