Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral
Abstract
:1. Introduction
2. Generalized Choquet Integral
- (1)
- ,
- (2)
- , and
- (3)
- if and . then .
- if X is μ-essentially bounded from above, then
- if X is μ-essentially bounded from below then
- (i)
- if X is bounded from below (above) then X is μ-essentially bounded from below (above) for every capacity μ on ; and,
- (ii)
- if, for some capacity μ on , X is μ-essentially bounded from below then
- (iii)
- if, for some capacity μ on , X is μ-essentially bounded from above, then
- (i)
- If is an -measurable function μ-essentially bounded from above, then so is . Furthermore,
- (ii)
- If is an -measurable function μ-essentially bounded from below, then so is . Moreover,
- (i)
- (ii)
- and
- (iii)
- (i)
- if and then
- (ii)
- if and then
- (iii)
- if then
- (iv)
- if then
3. Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral
4. Main Properties of the Mean
- (i)
- (ii)
- (iii)
- there exist and such thatand
- (i)
- (ii)
- (iii)
- there exist and , such thatand
- (i)
- (ii)
- (iii)
- one of the subsequent possibilities holds:
- (a)
- and there exist , such that
- (b)
- there exist with , such thatand
5. Conclusions
Funding
Conflicts of Interest
References
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Wójcik, S. Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral. Symmetry 2020, 12, 2104. https://doi.org/10.3390/sym12122104
Wójcik S. Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral. Symmetry. 2020; 12(12):2104. https://doi.org/10.3390/sym12122104
Chicago/Turabian StyleWójcik, Sebastian. 2020. "Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral" Symmetry 12, no. 12: 2104. https://doi.org/10.3390/sym12122104