For an increasing continuous function $f$ on $[0,1]$, a mod 1 transformation of $f$ is defined by $T_f(x)=f(x) \pmod 1$. In this talk, we consider the natural extension $\Sigma_f$ of the shift space induced by the mod 1 transformation $T_f$ satisfying some assumptions. We provide a necessary and sufficient condition under which any equilibrium measure for a potential function with bounded total oscillations on $\Sigma_f$ is a weak Gibbs measure.