a4d3aaf8   
   From: mailbox@dmitry-kazakov.de   
      
   On Fri, 28 Aug 2009 00:41:06 -0700 (PDT), un student wrote:   
      
   > On Aug 27, 9:13 pm, "Dmitry A. Kazakov"    
   > wrote:   
   >> On Thu, 27 Aug 2009 07:22:56 -0700 (PDT), un student wrote:   
   >>> A_a = \int_a^b f_a(t) dt =   
   >>>    { \int_a^b g(t) dt |   
   >>>       g(t) \in f_a(t) forall t in [a,b] }   
   > <..>   
   >> I guess if it was meant to be   
   >>   
   >> $\; >>\; A\_\backslash alpha\; = \; >>\; \backslash \{\; \backslash int\_a^b\; g(t)\; dt\; | \; >>\; g(t)\; \backslash leq\; f\_\backslash alpha(t)\; \backslash forall\; t\; \backslash in\; [a,b]\; \backslash \} \; >>$   
   >>   
   >> I .e. the alpha cut of the fuzzy integral is a set of plain integrals over   
   >> [a,b] computed for each function q dominated by the alpha-cut of f.   
   >   
   > I'm not sure if I still understand this. For example isn't alpha-cut a   
   > set?   
      
   Yes, it is.   
      
   > How come a function could be dominated by a set?   
      
   f_\alpha(t) is a fuzzy set for any t. You take all real-valued g's such   
   that for any t the outcome of g is in the support set of f_\alpha(t), i.e.   
      
   $\; f\_\backslash alpha(t)(g(t))\; \backslash ge\; 0 \;$   
      
   In other words, g may take any value from _\alpha(t). All these g's get   
   integrated on [a,b]. The results comprise some crisp set of "possible"   
   integrals. (It should be shown that with alpha increasing, the set get   
   narrower) Then the integral is proclaimed to have in x the truth value of   
   maximum of the alpha over sets of possible integrals containing x. (Zadeh's   
   extension principle).   
      
   >> P.P.S. There are many definitions of fuzzy integrals. Integral Sugeno comes   
   >> in mind, etc.   
   >   
   > What the previously presented definition is called? I tried to google   
   > up some resources but couldn't find anything but some research   
   > articles I can't get access to.   
      
   I am unsure if that has any name. Unless I am wrong it is merely a   
   (Lebesque?) integral extended on the case of fuzzy numbers.   
      
   Fuzzy numbers obey the inclusion rule. The idea is that you can split a   
   fuzzy number into a set of nested crisp sets by cutting it at different   
   levels (alpha). Then for a given alpha you do whatever operation (e.g.   
   integral) on all members of these crisp sets in all possible combinations   
   of (here follows some reasoning about set measures and countability, which   
   we ignore for the sake of clarity (:-)). The obtained crisp set of the   
   results for given alpha is then handled as I described above in order to   
   get a fuzzy set = fuzzy number of the extended operation.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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