In mathematics, the Sugeno integral, named after M. Sugeno, is a type of integral with respect to a fuzzy measure. Let

(X,\Omega)

be a

measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...

and let

h:X\to,1 /math> be an \Omega -

measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...

. The Sugeno integral over the

crisp set A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or ...

A \subseteq X

of the function

h

with respect to the fuzzy measure

g

is defined by: ::

\int_A h(x) \circ g 
=  \left min\left(\min_ h(x), g(A\cap E)\right)\right =  \left min\left(\alpha, g(A\cap F_\alpha)\right)\right

where

F_\alpha = \left\

. The Sugeno integral over the

fuzzy set In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a ...

\tilde

of the function

h

with respect to the fuzzy measure

g

is defined by: :

\circ g

where

h_A(x)

is the membership function of the fuzzy set

\tilde

Usage and Relationships

Sugeno integral is related to

h-index The ''h''-index is an author-level metric that measures both the productivity and citation impact of the publications, initially used for an individual scientist or scholar. The ''h''-index correlates with obvious success indicators such as winn ...

References

Gunther Schmidt Gunther Schmidt (born 1939, Rüdersdorf) is a German mathematician who works also in informatics. Life Schmidt began studying Mathematics in 1957 at Göttingen University. His academic teachers were in particular Kurt Reidemeister, Wilhelm Kl ...

(2006
Relational measures and integration

Lecture Notes in Computer Science ''Lecture Notes in Computer Science'' is a series of computer science books published by Springer Science+Business Media since 1973. Overview The series contains proceedings, post-proceedings, monographs, and Festschrifts. In addition, tutorials, ...

# 4136, pages 343−57, Springer books * M. Sugeno & T. Murofushi (1987) "Pseudo-additive measures and integrals",

Journal of Mathematical Analysis and Applications The ''Journal of Mathematical Analysis and Applications'' is an academic journal in mathematics, specializing in mathematical analysis and related topics in applied mathematics. It was founded in 1960, as part of a series of new journals on areas o ...

122: 197−222 {{mr, id=0874969 Fuzzy logic Measure theory Definitions of mathematical integration