If the minimum problem ($\mathcal{P}_{\infty}$) is the limit, in a variational sense, of a sequence of minimum problems with obstacles of the type \begin{equation*} \tag{$\mathcal{P}_{h}$} \min_{u\ge \phi_{h}} \int_{\Omega} \left[ f_{h}(x,Du) + a(x,u) \right] dx, \end{equation*} then ($\mathcal{P}_{\infty}$) can be written in the form \begin{equation} \tag{\mathcal{P}_{\infty}} \min_{u} \Big\{ \int_{\Omega} \left[ f_{\infty}(x,Du) + a(x,u) \right] dx + \int_{\bar{\Omega}} g_{\infty}(x,\bar{u}(x)) \, d\mu_{\infty}(x) \Big\} \end{equation} without any additional constraint.