In the present paper, we introduce the notions of lower envelope and upper envelope for a [0, $\infty$]-valued function $\mu$ defined on a proper sublattice $M$ of a locally complete $\sigma$-continuous lattice $L$, and we extend a finite-stable, supermodular usc-measure $\mu$ on a proper sublattice $M$ of $L$ to a quasi$^{*}$-measure (i.e., a supermodular usc-measure) on $L,$ which is $M_{\delta}$-inner regular. Analogously, we extend a submodular lsc-measure on $M$ to a quasi$_{*}$-measure (i.e., a submodular lsc-measure) on $L,$ which is $M_{\sigma}$-outer regular. Furthermore, we have studied notions of measuring envelopes in $D$-lattices in the context of null-additive, converse null-additive, superadditive and weak converse null-additive functions. 2000 Mathematics Subject Classification. 28A12, 28C15, 28B10