  In the paper [G.Pantsulaia, On Uniformly Distributed Sequences on [-1/2,1/2],
Georg. Inter. J. Sci. Tech.,4(3) (2013), 21--27], it was shown that
$\mu$-almost every element of $\mathbf{R}^{\infty}$ is uniformly distributed in
$[-\frac{1}{2}, \frac{1}{2}]$, where $\mu$ denotes Yamasaki-Kharazishvili
measure in $\mathbf{R}^{\infty}$ for which
$\mu([-\frac{1}{2},\frac{1}{2}]^{\infty})=1$. In the present paper the same set
is studying from the point of view of shyness and it is demonstrated that it is
shy in $\mathbf{R}^{\infty}$. In Solovay model, the set of all real valued
sequences uniformly distributed module 1 in $[-\frac{1}{2}, \frac{1}{2}]$ is
studied from the point of view of shyness and it is shown that it is the
prevalent set in $\mathbf{R}^{\infty}$.
