d1988151   
   From: mailbox@dmitry-kazakov.de   
      
   On Thu, 27 Aug 2009 07:22:56 -0700 (PDT), un student wrote:   
      
   > I'm having problems undestanding the fuzzy integral definition in my   
   > lecture notes. The definition is given by alpha-levels. First let f be   
   > a function from a certain interval to set F which is a set of certain   
   > kind of fuzzy numbers. That is f(t) is a function from R to [0,1].   
   >   
   > Let f_a(t) = [f(t)]_a i.e. the alpha levels of f(t)   
   >   
   > Fuzzy integral definition (by alpha-levels a)   
   >   
   > 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] }   
   >   
   > One thing I can't get is what means g(t) \in f_a(t)? How could a   
   > function be a member of a set? This notation remains a total mystery   
   > to me.   
      
   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.   
      
   P.S. Some work is required to show the premises:   
      
   1. the integrals of any such q exit   
   2. alpha-cuts are indeed nested, i.e. comprise a fuzzy subset of R   
      
   P.P.S. There are many definitions of fuzzy integrals. Integral Sugeno comes   
   in mind, etc.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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