Premeasure spaces, tight functions and extension toQuasi*-measure

The present paper deals with the theory of [0, 1] valued maps defined on a nonempty set X. We have concentrated over the study of two types of functions, viz. tight functions and smooth functions. The notions of lower and upper envelopes of a function β defined on a sublattice K of I^x are introduced, and are extensively used to prove several results. Finally it is obtained that every supermodular and smooth from above function can be extended to an inner regular quasi*-measure.