  In this paper we study the class $\mathcal{E}_{m}(\Omega)$ of $m-$subharmonic
functions introduced by Lu in \cite{L1}. We prove that the convergence in
$m-$capacity implies the convergence of the associated Hessian measure for
functions that belong to $\mathcal{E}_{m}(\Omega)$. Then we extend those
results to the class $\mathcal{E}_{m,\chi}(\Omega)$ that depends on a given
increasing real function $\chi$. A complete characterization of those classes
using the Hessian measure is given as well as a subextension theorem relative
to $\mathcal{E}_{m,\chi}(\Omega)$.
