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 ( mod 2 ) } ) = 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 .