Fuzzifying functions (Q581696)

Introducing the operations of addition \(\oplus\), multplication by a real number \(\odot\) and an operation of multiplication \(\cdot\) suitably on the set \(E=[-\infty,1)\subset R\) (suppose that the sets R and E are co- ordinated with regard to the relation of order), the author has considered \(E,\oplus,\odot),\) with the usual relation of order, as a ordered linear space and (E,\(\oplus,\cdot)\) as a field; the set (E,\(\oplus,\cdot)\) has been denoted by \(E_ a\) and a map \(\phi:\quad X\to E_ a\) as a real valued fuzzifying function where X is an arbitrary set. With the help of fuzzifying functions the author has introduced fuzzy sets, generalized fuzzy sets, distance between generalized fuzzy sets, the notion of \(\oplus\) measure, and Lebesgue integral; product of measures is also introduced with the help of fuzzifying functions ultimately leading to Fubini's theorem. A field of 'complex numbers' \(C_ a\) on which different Hilbert spaces could be built, is introduced. The possibility of defining fuzzy Hilbert spaces is considered.