Measurability and decomposition properties in the dual of a Riesz space. (English) Zbl 0780.46006

Filter, Wolfgang

Measure theory, Proc. Conf., Oberwolfach/Ger. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 28, 21-33 (1992).

[For the entire collection see Zbl 0747.00029.]
A Riesz space \(L\) is considered with separating order continuous dual and measurability of elements of the extended order continuous dual \(G(L)\) is defined. This approach is motivated by the measurability of real functions with respect to a \(\delta\)-ring \(D\). If \(A\in D\), \(M(D)\) is the Riesz space of all real-valued functions, then \(\hat f: M(D)\to R\) defined by \(\hat f(m)=\int f dm\), is a member of \(G(M(D))\). The property \(\{f\geq c\}\cap A\in D\) is translated into the language of \(G(L)\) by the representation theory of Riesz spaces. A Radon-Nikodym type theorem is proved.

Reviewer: B.Riečan (Bratislava)

MSC:

46A40	Ordered topological linear spaces, vector lattices
28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Keywords:

Riesz space; separating order continuous dual; measurability of elements of the extended order continuous dual; representation theory of Riesz spaces; Radon-Nikodym type theorem

Citations:

Zbl 0747.00029