Non-additive measure

Non-additive set functions, as for example outer measures and semi-variations of vector measures, appeared early in classical measure theory concerning countable additive set functions (cf. also Set function; Measure) or, more general, concerning finite additive set functions. The pioneer in the theory of non-additive set functions was G. Choquet [a3] with his theory of capacities (cf. also Capacity). This theory had influences on many parts of mathematics and different areas of sciences and technology ([a4], [a7], [a8], [a9], [a11], [a12]).

One can compare additive set functions (which are the basis for classical measure theory) and non-additive set functions in the following simple way. For a fixed set $A$ from a $\sigma$-algebra $\Sigma$, a classical measure $\mu : \Sigma \rightarrow [ 0 , + \infty ]$ gives that for every set $B$ from $\Sigma$ such that $A \cap B = \emptyset$ one has that $\mu ( A \cup B ) - \mu ( B )$ is always equal to a constant, $\mu ( A )$, i.e., it is independent of $B$. In contrast, for a non-additive set function $m$ the difference $m ( A \cup B ) - m ( B )$ depends on $B$ and can be interpreted as the effect of $A$ joining $B$.

Non-additive set functions are extensively used in decision theory ([a7], [a12]), mathematical economy, social choice problems, with early traces by R.J. Aumann and L.S. Shapley in [a2]. Many authors have investigated various kinds of non-additive set functions, such as subadditive and superadditive set functions, submeasures, $k$-triangular set functions, $t$-conorm and pseudo-addition decomposable measures (cf., e.g., Idempotent analysis), null-additive set functions, and many other types. Although in many results the monotonicity of the observed set functions was supposed, there are some results concerning certain classes of set functions which include also non-monotone set functions (for example superadditive set functions, $k$-triangular set functions; [a2], [a9]).

On the other hand, "fuzzy measures" regarded as monotone and continuous set functions were investigated by M. Sugeno in [a10] with the purpose to evaluate non-additive quantities in systems engineering. This notion of "fuzziness" is different from the one given by L.A. Zadeh (cf. also Non-precise data). Namely, instead of taking membership grades of a set, one takes (in the fuzzy measure approach) the measure that a given unlocated element belongs to a set.

Many important types of non-additive set functions occur in various branches of mathematics, such as potential theory ([a3]), harmonic analysis, fractal geometry ([a6]), functional analysis, the theory of nonlinear differential equations, the theory of difference equations, and in optimization ([a9], [a12]).

Also, interest in non-additive set functions is growing. In artificial intelligence, belief functions have been applied to model uncertainty, [a12]. Belief functions, corresponding plausibility measures and other kinds of non-additive set functions are used in statistics, [a8]. Non-additive expected utility theory has been applied, for example, in multi-stage decision and economics, [a7].

The unification of many different kinds of non-additive set functions is achieved by the wide class of so-called null-additive set functions. A set function $m$ defined on a family $\mathcal{D}$ of sets such that it is closed with respect to the union and $\emptyset \in \cal D$, and with values in $[ - \infty , \infty ]$ (or more general in a semi-group with a neutral element $0$) and such that $m ( B ) = 0$ implies $m ( A \cup B ) = m ( A )$ whenever $A , B \in \Sigma$, $A \cap B = \emptyset$, is called a null-additive set function. These set functions have very nice properties with respect to the usual notions in measure theory and occur naturally in the theories of integrals ([a9], [a12]). The origins of null-additive set functions are in papers of V.N. Aleksyuk [a1] (under the name "quasi-measures" ) and I. Dobrakov [a5] (under the name "submeasures" ).

References

[a1]	V.N. Aleksjuk, "Two theorems on existence of quasibase for a family of quasimeasures" Izv. Visch. Uchebn. Zav. , 6 : 73 (1968) pp. 11–18 (In Russian)
[a2]	R.J. Aumann, L.S. Shapley, "Values of non-atomic games" , Princeton Univ. Press (1974)
[a3]	G. Choquet, "Theory of capacities" Ann. Inst. Fourier (Grenoble) , 5 (1953) pp. 131–295
[a4]	D. Denneberg, "Non-additive measure and integral" , Kluwer Acad. Publ. (1994)
[a5]	I. Dobrakov, "On submeasures I" Diss. Math. , 112 (1974)
[a6]	K. Falconer, "Fractal geometry" , Wiley (1990)
[a7]	M. Grabisch, H.T. Nguyen, E.A. Walker, "Fundamentals of uncertainity calculi with application to fuzzy inference" , Kluwer Acad. Publ. (1995)
[a8]	P.J. Huber, "The use of Choquet capacities in statistics" Bull. Int. Inst. Statist. , 45 (1973) pp. 181–191
[a9]	E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. /Ister (1995)
[a10]	M. Sugeno, "Theory of fuzzy integrals and its applications" PhD Thesis Tokyo Inst. Technol. (1974)
[a11]	M. Sugeno, T. Murofushi, "Pseudo-additive measures and integrals" J. Math. Anal. Appl. , 122 (1987) pp. 197–222
[a12]	Z. Wang, and G. Klir, "Fuzzy measure theory" , Plenum (1992)