On a generalized infimal convolution of set functions
Abstract
Having in mind the ideas of J. Moreau, T. Strömberg and Á. Száz, for any function f and g of one power set 𝓟(X) to another 𝓟(Y), we define an other function (f⚹g) of 𝓟(X) to 𝓟(Y ) such that
(f⚹g)(A) = ∩{f(U)∪g(V): A⊂U∪V⊂X}
for all A⊂X. Thus (f⚹g) is a generalized infimal convolution of f and g.
We show that if f and g preserve arbitrary unions, then (f⚹g) also preserves arbitrary unions. Moreover, if F and G are relations on X to Y such that
F(x) = f({x}) and G(x) = g({x})
for all x∈X, then
(f⚹g)(A) = F⋂G[A]
for all A⊂X.
Keywords
infimal convolution; union-preserving set functions
References
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Department of Mathematical Analysis, Budapest University of Technology and Economics, Hungary Hungary
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