Finding a measure, a ring theoretical approach
DOI:
https://doi.org/10.14419/ijamr.v6i1.6864Published:
2016-12-07Keywords:
Boolean ring, Measure space, Pseudometric space, Uniform probability space.Abstract
Let \(S\) be a nonempty set and \(F\) consists of all \(Z_{2}\) characteristic functions defined on \(S\). We are supposed to introduce a ring isomorphic to \((P(S),\triangle,\cap)\), whose set is \(F\). Then, assuming a finitely additive function $m$ defined on \(P(S)\), we change \(P(S)\) to a pseudometric space \((P(S),d_{m})\) in which its pseudometric is defined by \(m\). Among other things, we investigate the concepts of convergence and continuity in the induced pseudometric space. Moreover, a theorem on the measure of some kinds of elements in \((P(S),m)\) will be established. At the end, as an application in probability theory, the probability of some events in the space of permutations with uniform probability will be determined. Some illustrative examples are included to show the usefulness and applicability of results.
References
[1] R. Abraham, J. E. Marsden, T. Ratiu, Manifolds, tensor analysis, and applications, Springer Verlag, (1988).
[2] J. M. Almira, “Yet another application of the Gauss-Bonnet theorem for the sphereâ€, Bulletin of the Belgian Mathematical Society, V (14), (2007), 341-342.
[3] J. M. Almira, A. Romero, “Some Riemannian geometric proofs of the fundamental theorem of algebraâ€, Differential Geometry-Dynamical Systems, V (14), (2012), 1-4.
[4] R. V. Churchill, J. W. Brown, R. F. Verhey, Complex variables and applications, Mc Graw-Hill, (2004).
[5] O. R. B. de Oliveira, “The fundamental theorem of algebra: from the four basic operationsâ€, American Mathematical Monthly (119), V (9), (2012), 753-758.
[6] M. Eisermann,“The fundamental theorem of algebra made effective: An elementary real-algebraic proof via Sturm chainsâ€, American Mathematical Monthly, (119), V (9), (2012), 715-752.
[7] G. B. Folland, Real analysis, modern techniques and applications, John Wiley & Sons, Inc., (1999).
[8] S. Givant, P. Halmos, Introduction to Boolean algebra, Springer Verlag, (2009).
[9] V. Guillman, A. Pollack, Differential topology, AMS Chelsea Publishing, (2010).
[10] M. W. Hirsch, S. Smale, R. L. Devaney, Differential equations, dynamical systems, and an introduction to chaos, Academic Press, (2013).
[11] J. W. Milnor, Topology from the differentiable view point, Princeton University Press, (1997).
[12] A. R. Schep, “A simple complex analysis and an advanced calculus proof of the fundamental theorem of algebraâ€, American Mathematical Monthly, V (116), (2009), 67-68.
[13] J. Stewart, Calculus, Brooks/Cole, (2012).
[14] R. E. Walpole, R. H. Myers, S. L. Myers, K. Ye, Probability and statistics for engineers and scientists, Prentice Hall, (2011).
View Full Article:
License
Authors who publish with this journal agree to the following terms:
                          [1]           Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
                          [2]           Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
                          [3]           Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).