J. A. Chatfield, "A representation theorem for operators on a space of interval functions", International Journal of Mathematics and Mathematical Sciences, vol. 1, Article ID 589151, 12 pages, 1978. https://doi.org/10.1155/S0161171278000319
A representation theorem for operators on a space of interval functions
Suppose is a Banach space of norm and is the set of real numbers. All integrals used are of the subdivision-refinement type. The main theorem [Theorem 3] gives a representation of where is a function from to such that , , , and each exist for each and is a bounded linear operator on the space of all such functions . In particular we show that where each of , , and depend only on , is of bounded variation, and are except at a countable number of points, is a function from to depending on and denotes the points in . for which or . We also define an interior interval function integral and give a relationship between it and the standard interval function integral.
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