Finding a measure, a ring theoretical approach
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2016-12-07 https://doi.org/10.14419/ijamr.v6i1.6864
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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.
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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).
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How to Cite
Parsian, A. (2016). Finding a measure, a ring theoretical approach. International Journal of Applied Mathematical Research, 6(1), 1-3. https://doi.org/10.14419/ijamr.v6i1.6864Received date: 2016-10-12
Accepted date: 2016-11-17
Published date: 2016-12-07