The Radon-Nikodým theorem in bornological spaces. (Spanish: El teorema de Radon-Nikodym en espacios bornológicos).

Abstract

The author presents some Radon-Nikodým theorems—i.e., if (Ω,Σ,μ) is a finite measurable space and m is a μ -continuous vector measure then {m(A)/μ(A):A∈Σ} being compact in some sense implies that m is an integral. Theorem 6: One has a Radon-Nikodým theorem for Fréchet spaces. Theorem 8: One has a Radon-Nikodým theorem for vector measures of finite variation and for a quasicomplete space E with the metrizable Pietsch property. (E is said to have the metrizable Pietsch property if X⊂E N with ∑p(x n )<∞ for each (x n )∈X and each continuous seminorm p implies the existence of a B , B⊂E , absolutely convex, bounded and metrizable, with gauge P B , and such that ∑P B (y n )≤1 for each (y n )∈X .) G. Y. H. Chi proved this result [Measure theory (Oberwolfach, 1975), pp. 199–210, Lecture Notes in Math., 541, Springer, Berlin, 1976; with a compactness hypothesis for m(A)/μ(A) , instead of weak compactness. Unfortunately the proofs are not quite clear or even quite exact. In the reference to the Grothendieck book there is a B instead of an 8.