Abstract
In this note, the convergence of the sum of two convergent sequences of measurable functions is studied by means of two types of absolute continuity of fuzzy measures, i.e., strong absolute continuity of Type I, and Type VI. The discussions of convergence a.e. and convergence in measure are done in the general framework relating to a pair of monotone measures, and general results are shown. The previous related results are generalized.
Highlights
In fuzzy measure and fuzzy integral theory, many results in the classical measure theory no longer hold generally without additional conditions for fuzzy measures
We have shown the equivalences between the convergence (a.e. or in measure) of the sum of two convergent sequences of measurable functions and several types of absolute continuity of fuzzy measures
I and Type VI of fuzzy measures have been described by using convergence of sequence of measurable functions
Summary
In fuzzy measure and fuzzy integral theory, many results in the classical measure theory no longer hold generally without additional conditions for fuzzy measures. For a fuzzy measure μ, in general, the above Equations (1) and (2) may not be true. The above Equations (1) and (2) were generalized to fuzzy measure spaces under the conditions of weak null-additivity and pseudometric generating property of set functions [3]. We consider a pair of fuzzy measures λ and ν defined on the same measurable space ( X, A). Under the condition of strong absolute continuity of Type I Type VI), we obtain the following result:. Comparing Equation (3) with Equation (1), and Equation (4) with Equation (2), respectively, we see that the general results are obtained in the framework concerning a pair of monotone measures.
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