Integrability and closest approximation representations
- 16 Downloads
Suppose U is a set,F is a field of subsets of U, andp AB is the set of all real-valued bounded finitely additive functions defined onF. This paper consists of two main parts. In the first, a previously given (seeRiv. Math. Univ. Parma, (3),2(1973), pp. 251–276) notion of a subset ofp AB defined by certain closure properties and called a C-set, is considered, and those C-sets that are linear spaces are characterized. Now, suppose γ is a function whose domain isF and whose range is a collection of number sets with bounded union. The set,Jγ, of all elements ofp AB with respect to which γ is integrable, for refinements of subdivisions, is a C-set and a linear space (seeRend. Sem. Mat. Univ. Padova,52 (1974), pp. 1–24). The second part of this paper concerns, for μ inp AB and nonnegative-valued, a representation of the element ofJγ closest to μ with respect to variation norm.
KeywordsMain Part Linear Space Variation Norm Additive Function Approximation Representation
- E. Hellinger,Die Orthogonalinvarianten quadratischer Formen von unendlichvielen Variablen, Diss. Gottingen (1907).Google Scholar