  In this paper, we introduce the class $\mathcal{E}_{m,F}(\Omega)$ and solve
complex $m$-Hessian equations in the class $\mathcal{E}_{m,F}(\Omega)$.
Afterthat, we study subextension in the class $\mathcal{E}_{m,F}(\Omega)$ with
the weighted Hessian measure of subextension unchanged. This is an extensive
version of the result in [24] in the case when the smaller domain is relatively
compact inside the large one.
