Weakly tight functions and their decomposition

The notion of a signed measure arises if a measure is allowed to take on both positive and negative values. A set that is both positive and negative with respect to a signed measure is termed as a null set. Some concepts in measure theory can be generalized by means of classes of null sets. An abstract formulation and proof of the Lebesgue decomposition theorem using the concept of null sets is given by Ficker [5]. A real-valued function satisfying certain properties that can be expressed as a difference of two nonnegative functions possessing the same properties is called “decomposable.” Several Jordan-decompositiontype theorems are exhibited in [3]. Faires and Morrison [4] exposed conditions on a vector-valued measure that ensure vector-valued Jordan-decomposition-type theorem to hold. For a signed null-additive fuzzy measure, a Jordan-decomposition-type theorem is investigated by Pap in [11]. The problem of generation of measures by tight functions defined on a lattice of sets has been taken up by several authors [1, 2, 6, 8, 9]. Nayak and Srinivasan [10] initiated a weaker form of tightness for a real-valued function μ defined on a lattice of sets to decompose μ as a difference μ+ −μ− and then extended it to a countably additive measure. In Section 2, we have defined and studied the notions of measuring envelopes, modular functions, and additive functions. The notions of superadditive and subadditive functions are also given with the help of pointwise addition of elements in IX . The lower envelope β∗ of a superadditive function β defined on a sublattice K of IX turns out to be superadditive. In Section 3, we introduce the notion of a weakly tight function β : K →R, where K is a sublattice of IX containing 0 and 1 (cf. [10]). The condition imposed on the [0,1]-valued function β to be a weakly tight function is less restrictive than that for being a tight function. It is proved that a superadditive, monotone, and weakly tight function