Weighted $m$-subharmonic measure and $(m, \psi)$-capacity ${{C}_{m}} (K,D,\psi)$ of a condenser $(K,D)$

We introduce the concept of the weighted $m$-subharmonic measure and the capacity of a condenser $(K,D)$ with a weighted function $\psi(z)\in C (K)$ for a compact set $K\subset D$, where $D\subset {{\mathbb{C}}^{n}}$ is a domain, in the class of $m$-subharmonic functions.