For an open continuous mapping \(\pi\:X\to\Pi\), the author proves the existence of a majorizable lifting of the space of quasi-Radon measures defined on the Borel \(\sigma\)-algebra of a locally compact paracompact space \(\Pi\) to the space of quasi-Radon measures defined on the Borel \(\sigma\)-algebra of a locally compact space \(X\). This result implies a theorem on extension of majorizable mappings defined on a closed subgroup of a locally compact Abelian group.