Approximation of capacities
by
Inna Hlushak
Precarpathian National University
Coauthors: Oleg Nykyforchyn

Given a capacity c on a metric compactum X we investigate optimal approximations by ∪-capacities (or ∩-capacities) and optimal approximations by capacities on a closed subspace X₀ ⊂ X.

On the space MX of upper semicontinuous capacities [3] on a metric compactum (X, d) we consider the following metric :

dM(c, c')= inf {e > 0 | c(Oe(F))+e ≥ c'(F), c'(Oe(F))+e ≥ c(F) for any closed subset F ⊂ X}

where O_e(F) stands for the closed e-neighborhood of F in X.

We call a capacity c ∈ MX a ∩-capacity (necessity measure) [2] if for all closed subsets A, B of X we have c(A∩B)=min{c(A), c(B)}. Analogously, we call a capacity c ∈ MX a ∪-capacity (also possibility measure [2]), if c(A∪B)=max{c(A), c(B)} for all closed sets A, B ⊂ X.

We denote the set of all ∩-capacities (∪-capacities) on a compactum X by M_∩X ( M_∪X). It is proved [2, 3], that the space MX and its subspaces M_∪X and M_∩X are compacta.

If c is a capacity on X, then the function kX(c) that is determined on the set of all closed subsets X by the formula kX(c)(F)=1-c(X\F) is a capacity as well. It is called the dual or conjugate to c. It is also proved in [2] that a capacity c on a compactum X is a ∩-capacity if and only if the dual capacity kX(c) is a ∪-capacity.

Each capacity c₀ on a closed subspace X₀ of a space X we extend to X by the formula c(F)=c₀(F∩X₀) for closed subsets F ⊂ X.

We show that distance from a capacity c to the subspace M_∪X is equal to the infimum of all e ≥ 0 such that the following condition (*) holds:

c(X\OeBae) ≤ a+e for a ∈ I, where Bae={x ∈ X | c(Oe(x)) ≥ a-e}

Here is an algorithm that finds one of ∪-capacities that are nearest to c : we find a least e ≥ 0 such that (*) is valid and use the obtained sets B^a_e to construct a capacity c₀ ∈ M_∪X that is an optimal approximation (in fact it is the greatest of optimal approximations). The capacity c₀ is of the form:

c0(F) = sup {a ∈ I | F∩Bae ≠ ∅}

for any closed subset F of X.

The map kX is an isometry of (MX, d_M) onto itself, therefore the capacity c'₀ ∈ M_∪X that is nearest to a capacity c ∈ MX can be found as follows : we find the capacity c₀ ∈ M_∪X which is nearest to kX(c) and than we go to the dual capacity : c'₀=kX(c₀).

It is also proved that the distance from a capacity c on X to a subspace MX₀ ⊂ MX is equal to e = min{e ≥ 0|c(O_e(X₀)) ≥ 1-e, c(X\O_e(X₀)) ≤ e}. One of nearest to c ∈ MX capacities on X₀ is defined by the formula:

ce = min (c(Oe(F))+e, 1), F ≠ ∅, 0, F=∅.

A capacity c₀ ∈ MX₀ is one of nearest to c if and only if

kX(kX(c)e) ≤ c0 ≤ ce.

References

[1] Choquet G. Theory of Capacity, Ann. l'Institute Fourier, 5 (1953-1954), 131-295

[2] Hlushak I.D., Nykyforchyn O.R., Submonads of the capacity monad, 2007, Submitted to Carpathian Journal of Mathematics

[3] Zarichnyi M.M., Nykyforchyn O.R., Functor of capacities in the catagory of compacta, Matem. sb., 2008, 199:2, 3-26 (in Russian)

Date received: April 30, 2008