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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202007494 |
| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |