On some classes of distance distribution function-valued submeasures

On distance distribution functions-valued submeasures related to aggregation functions

“…Thus, the triple (Σ, ρ, τ L,T ) is an L-Menger probabilistic pseudo-metric space, see [11,Theorem 3.2]. Since for…”

“…In our previous papers [12] and [11] we have introduced and investigated submeasure notions related to probabilistic metric spaces (PM-spaces, for short), see [16]. Our considerations of a submeasure notion in paper [12] were closely related to the Menger PM-space (Ω, F , τ T ) where τ T is the triangle function in the form…”

“…Note that similar considerations were introduced and discussed in the framework of probabilistic metric spaces, see for example the monograph [8]. For such reasons in paper [11] we have provided a generalization of τ T -submeasures which involves suitable operations L replacing the standard addition + on R + such that the underlying function (1) is a triangle function and thus the underlying space is the so-called L-Menger PM-space. Since t-norms are rather special operations on the unit interval [0, 1], we have also mentioned few possible generalizations of a submeasure notion based on aggregation operators and convolution of distance distribution functions, i.e., such submeasures which can be used in non-Menger PM-spaces (e.g., in the Wald spaces), but also in wider class of PM-spaces.…”

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 .…”

“…The study of probabilistic-valued set functions continued in papers [6] and [7], where a more general concept has been proposed. Motivated by the paper of Shen [20] we have recently introduced in [5] a general framework of τ -decomposable set functions which covers all the mentioned approaches and treat them in a unified way.…”