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...

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Main Authors:	Robert Gardner, Robert Price
Format:	Article
Language:	English
Published:	Hindawi Limited 2002-01-01
Series:	International Journal of Mathematics and Mathematical Sciences
Online Access:	http://dx.doi.org/10.1155/S0161171202007494

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spelling	doaj-58590c4387194ec184418c99c9d76dc02020-11-24T22:54:29ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-01291058558910.1155/S0161171202007494Translation invariance and finite additivity in a probability measure on the natural numbersRobert Gardner0Robert Price1Department of Mathematics, Box 70663, East Tennessee State University, Johnson City 37614, TN, USADepartment of Mathematics, Box 70663, East Tennessee State University, Johnson City 37614, TN, USAInspired 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.http://dx.doi.org/10.1155/S0161171202007494
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Robert Gardner Robert Price
spellingShingle	Robert Gardner Robert Price Translation invariance and finite additivity in a probability measure on the natural numbers International Journal of Mathematics and Mathematical Sciences
author_facet	Robert Gardner Robert Price
author_sort	Robert Gardner
title	Translation invariance and finite additivity in a probability measure on the natural numbers
title_short	Translation invariance and finite additivity in a probability measure on the natural numbers
title_full	Translation invariance and finite additivity in a probability measure on the natural numbers
title_fullStr	Translation invariance and finite additivity in a probability measure on the natural numbers
title_full_unstemmed	Translation invariance and finite additivity in a probability measure on the natural numbers
title_sort	translation invariance and finite additivity in a probability measure on the natural numbers
publisher	Hindawi Limited
series	International Journal of Mathematics and Mathematical Sciences
issn	0161-1712 1687-0425
publishDate	2002-01-01
description	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.
url	http://dx.doi.org/10.1155/S0161171202007494
work_keys_str_mv	AT robertgardner translationinvarianceandfiniteadditivityinaprobabilitymeasureonthenaturalnumbers AT robertprice translationinvarianceandfiniteadditivityinaprobabilitymeasureonthenaturalnumbers
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Translation invariance and finite additivity in a probability measure on the natural numbers

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