On the inner Daniell-Stone and Riesz representation theorems. (English) Zbl 0956.28012

König, Heinz

Doc. Math. 5, 301-315 (2000).

The author is concerned with integral representations of an isotone positive-linear functional \(I\) defined as a Stonean lattice cone \(E\subset [0,\infty[^X\), where \(X\) is a nonempty set, with values in \([0,\infty[\). The results complement the material contained in Chapter V of the author’s book [“Measure and integration. An advanced course in basic procedures and applications” (1997; Zbl 0887.28001)], cited below as MI. The main novelity of the paper under review is the elimination of the previous tightness (= inner regularity) conditions on \(I\) at the cost of introducing some additional assumptions on \(E\). Those assumptions are of the form: \(v-u\in F\) for all \(u,v\in E\) with \(u\leq v\), where \(F\subset [0,\infty[^X\) is a certain hull of \(E\), e.g., \(F= E_\sigma\) or \(E_\tau\), the family of pointwise infima of countable, resp., arbitrary subsets of \(E\). In the special case where \(X\) is a Hausdorff topological space and the elements of \(E\) are upper semicontinuous functions each vanishing outside a compact subset of \(X\), Riesz type representation theorems are derived. A further specialization yields one of the versions of the classical Riesz representation theorem for locally compact \(X\). The proofs of the main results strongly appeal to MI as well as to a subsequent paper by the author [Ann. Univ. Sarav., Ser. Math. 9, No. 2, 123-153 (1998; Zbl 0933.28001)]. The paper under review also contains some illuminating remarks on recent work of V. K. Zakharov and A. V. Mikhalëv [see, e.g., Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 5, 37-82 (1999)] concerning what those authors call the problem of the general Radon representation for a Hausdorff space.
For the reader’s convenience we note that: (1) The functional \(I\) above is called an “elementary integral” in MI. (2) The present notation “\(\geq(E)\)” replaces “\({\mathfrak T}(E)\)” of MI. (3) The functional \(I\) satisfying condition 3) of Theorem 1.3, which also appears in Theorem 3.7, is called “tight” in MI. (4) The term “rich” of MI is now changed to “\(\tau\) rich”.

Reviewer: Zbigniew Lipecki (Wrocław)

MSC:

28C05	Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28A12	Contents, measures, outer measures, capacities
28A25	Integration with respect to measures and other set functions
28C15	Set functions and measures on topological spaces (regularity of measures, etc.)

Keywords:

inner extensions; inner premeasures; Radon measures; Radon premeasures; positive (isotone) linear functionals; inner sources; inner preintegrals; Radon preintegrals; tightness; theorem of Kisyński

Citations:

Zbl 0887.28001; Zbl 0933.28001