This paper introduces concepts of the σ-possibility measure and the CP-system and ECP-system on a class of fuzzy sets, and shows that such a σ-possibility measure must be a semi-continuous fuzzy measure. Furthermore, a necessary and sufficient condition to extend a σ-possibility measure on a class of fuzzy sets is established, and some extension theorems of such σ-possibility measures are proven. Therefore, the extension problem of a class of semi-continuous fuzzy measures is solved.