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∈□| 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 □. Finally, a resolution to the two envelopes exchange paradox is presented in terms of m.
Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.
Gardner, Robert; and Price, Robert. 2002. Translation Invariance and Finite Additivity in a Probability Measure on the Natural Numbers. International Journal of Mathematics and Mathematical Sciences. Vol.29(10). 585-589. https://doi.org/10.1155/S0161171202007494 ISSN: 0161-1712