On a Certain Class of Submeasures Based on Triangular Norms

Abstract

In this paper we study a generalization of a submeasure notion which is related to a probabilistic concept, especially to Menger spaces where triangular norms play a crucial role. The resulting notion of a τ T -submeasure is suitable for modeling those situations in which we have only probabilistic information about the measure of the set. We characterize a class of universal τ T -submeasures (i.e., τ T -submeasures for an arbitrary t-norm T ) and give explicit formulas for τ T -submeasures for some classes of t-norms. Also, transformations and aggregations of τ T -submeasures are discussed.

On distance distribution functions-valued submeasures related to aggregation functions

“…If L = K 1 , then its index is usually omitted, and we simply speak about τ Asubmeasure. Clearly, for A = T (a left-continuous t-norm), and L = K 1 the τ L,A -submeasure reduces to τ T -submeasure from [12]. For instance, for L = K ∞ we get a τ max,T -submeasure related to a non-Archimedean Menger PM-space (Ω, F , τ max,T ).…”

“…Hence, for a = b = 3 2 c, we have γ ∈ Θ Π , but γ / ∈ Θ M . Observe also that γ is not related to any numerical submeasure, see [12]. Example 2.4 For a positive real number p consider the class M p ⊂ A which is usually called the p-mean (or, the Hölder mean) and is defined as…”

“…As it is already known, see [12,Theorem 1], to each numerical submeasure η on Σ corresponds γ ∈ Θ L,M , L ≥ K 1 , in the form…”

“…with T being a left-continuous t-norm. The associated submeasure notion was defined as follows, see [12,Definition 3] and Section 2 for necessary notations. (c) γ E∪F (x + y) ≥ T (γ E (x), γ F (y)), x, y > 0, E, F ∈ Σ, is said to be a τ T -submeasure.…”

An integral with respect to probabilistic-valued decomposable measures

“…Also, monotonicity of τ comes into play to provide monotonicity of the set function γ. We summarize that the probabilistic-valued (sub)measures defined and studied in [2], [6], [7], [10] and [20] are only special cases of τ -decomposable set functions w.r.t. a different choice of triangle function τ .…”