On a generalized infimal convolution of set functions

Gergely Pataki

Published: 2013-09-30

Vol. 27 (2013)

On a generalized infimal convolution of set functions

Gergely Pataki

Abstract

Having in mind the ideas of J. Moreau, T. Strömberg and Á. Száz, for any function f and g of one power set ????(X) to another ????(Y), we define an other function (f⚹g) of ????(X) to ????(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

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Pataki, G. (2013). On a generalized infimal convolution of set functions. Annales Mathematicae Silesianae, 27, 99–106. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14007

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References

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Vol. 27 (2013)
Published: 2013-09-30

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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