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The \(M\)-components of level sets of continuous functions in WBV. (English) Zbl 0991.54013

Author’s abstract: “It is proved that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of \(M\)-connected components of its level sets, coincides when the function is a continuous function in WVB. Both function spaces are frequently used as models for images. Thus, if the domain \(\text{cl}(\Omega)\) of the image is a Jordan domain, a rectangle, for instance, and the image \(u \in C(\text{cl}(\Omega)) \cap \text{WBV}(\Omega)\) (being constant near \(\delta \Omega \)), we prove that for almost all levels \( \lambda \) of \(u\), the classical connected components of positive measure of \([u \geq \lambda ]\) coincide with the \(M\)-components of \([u \geq \lambda ]\). Thus the notion of \(M\)-component can be seen as a relaxation of the classical notion of connected component when going from \(C(\text{cl}(\Omega))\) to \(\text{WBV} (\Omega)\)”.

MSC:

54C30 Real-valued functions in general topology
26A45 Functions of bounded variation, generalizations
28A99 Classical measure theory
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