  We consider a generalized Cantor set $E(\omega)$ for an infinite sequence
$\omega=(q_n)_{n=1}^{\infty}\in (0, 1)^{\mathbb N}$, and consider the moduli
space $M(\omega)$ for $\omega$ which are the set of $\omega'$ for which
$E(\omega')$ is conformally equivalent to $E(\omega)$.
  In this paper, we may give a necessary and sufficient condition for
$D(\omega):=\mathbb C\setminus E(\omega)$ to be a uniform domain. As a
byproduct, we give a condition for $E(\omega)$ to belong to $M(\omega_0)$, the
moduli space of the standard middle one-third Cantor set. We also show that the
volume of the moduli space $M(\omega)$ with respect to the standard product
measure on $(0, 1)^{\mathbb N}$ vanishes under a certain condition for
$\omega$.
