In this paper we treat the problem of connection between the convergence in $m-$capacity and the convergence of the Hessian measure for a sequence $f_j$ of $m-$subharmonic functions. We prove first that, under some conditions, the convergence of $f_j$ in capacity $Ca{p}_m$ implies the weak convergence of the Hessian measures $H_m(f_j)$. Then we show that the converse sense of convergence is also true in some particular cases.

This paper investigated the properties of maximal subextensions for $m$-subharmonic ( $m$-sh) functions in the context of complex Hessian operators. We established three main theorems that significantly advance the theory. First, we provided a complete characterization of maximal subextensions in the class ${\mathcal{F}}_m^a(\mathrm{\Omega })$, showing that any subextension satisfying a minimality condition on its Hessian mass must coincide with the maximal subextension. Second, we proved that for functions in ${\mathcal{E}}_m(\mathrm{\Omega })$, the maximal subextension relates to the original function through an $m$-maximal function and preserves Lelong numbers. Third, we demonstrated convergence stability, proving that for sequences with boundary values in ${\mathcal{F}}_m(\mathrm{\Omega },f)$ converging in $L_{\text{loc}}^1$ and uniformly bounded below, their maximal subextensions converged to the maximal subextension of the limit function.