  We consider the random $\beta$-transformation $K_{\beta}$, defined on
$\{0,1\}^{\mathbb N}\times[0, \frac{\lfloor\beta\rfloor}{\beta-1}]$, that
generates all possible expansions of the form
$x=\sum_{i=0}^{\infty}\frac{a_i}{\beta^i}$, where $a_i\in
\{0,1,\cdots,\lfloor\beta\rfloor\}$. This transformation was first introduced
by Dajani and Kraaikamp, and later studied by Dajani and de Vries, where two
natural invariant ergodic measures were found. The first is the unique measure
of maximal entropy, and the second is a measure of the form $m_p\times
\mu_{\beta}$, with $m_p$ the Bernoulli $(p,1-p)$ product measure and
$\mu_{\beta}$ is a measure equivalent to Lebesgue measure. In this paper, we
give an uncountable family of $K_{\beta}$-invariant exact $g$-measures for a
certain collection of algebraic $\beta$'s.
