Structure of All Real-Valued Sequences Uniformly Distributed in $$[-\frac{1}{2}, \frac{1}{2}]$$ [ - 1 2, 1 2 ] from the Point of View of Shyness

Gogi Pantsulaia ()
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Gogi Pantsulaia: Georgian Technical University, Department of Mathematics

Chapter Chapter 3 in Applications of Measure Theory to Statistics, 2016, pp 47-56 from Springer

Abstract: Abstract In [P5], it was shown that $$\mu $$ μ almost every element of $$\mathbf {R}^{\infty }$$ R ∞ is uniformly distributed in $$[-\frac{{1}}{{2}}, \frac{{1}}{{2}}]$$ [ - 1 2 , 1 2 ] , where $$\mu $$ μ denotes the Moore–Yamasaki–KharazishviliMoore–Yamasaki–Kharazishvili type measure measure in $$\mathbf {R}^{\infty }$$ R ∞ for which $$\mu ([-\frac{1}{2},\frac{1}{2}]^{\infty }) = 1$$ μ ( [ - 1 2 , 1 2 ] ∞ ) = 1 . In the present chapter the same set is studied from the point of view of shyness and it is demonstrated that it is shy in $$\mathbf {R}^{\infty }$$ R ∞ . In the Solovay modelSolovay Model (SM) , the structure of the set of all sequences uniformly distributed modulo 1Uniformly distributed modulo 1 in $$[-\frac{{1}}{{2}}, \frac{{1}}{{2}}]$$ [ - 1 2 , 1 2 ] is studied from the point of view of shyness and it is shown that it is the prevalent setPrevalent set in $$\mathbf {R}^{\infty }$$ R ∞ .

Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-45578-5_3

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DOI: 10.1007/978-3-319-45578-5_3

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