Positivity 12 (2008), 105–118
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010105-14, published online October 29, 2007
New Version of the Daniell-Stone-Riesz
Dedicated to the Memory of Helmut H. Schaefer
Abstract. The traditional representation theorems after Daniell-Stone and
Riesz were in a kind of separate existence until Pollard-Topsøe 1975 and
Topsøe 1976 were the ﬁrst to put them under common roofs. In the same
spirit the present article wants to obtain a uniﬁed representation theorem in
the context of the author’s work in measure and integration. It is an inner
theorem like the previous ones. The basis is the recent comprehensive inner
Daniell-Stone theorem, so that in particular there are no a priori assumptions
on the additive behaviour of the data.
Mathematics Subject Classiﬁcation (2000). 28-02; 28C05; 28C15.
Keywords. Daniell-Stone and Riesz representation theorems, Stonean lattices
and Stonean lattice cones, Inner premeasures, Inner extensions, Inner enve-
lopes, Choquet integral, Set-theoretical compactness.
The present article wants to put an adequate uniﬁed result on top of the collec-
tion of representation theorems of Daniell-Stone and Riesz type obtained in the
context of the author’s work in measure and integration described in [12, 15]. We
shall concentrate on the inner development which turned out to be more profound
than the outer one. We recall that its basic concepts are the inner • premeasures
ϑ : S → [0, ∞[ on a lattice S with ∅ ∈ S in a nonvoid set X and their inner
• extensions (• = στ with = ﬁnite, σ = sequential, τ = nonsequential), and
that its basic devices are the inner • envelopes ϑ
: P(X) → [0, ∞] of the isotone
set functions ϑ : S → [0, ∞[ with ϑ(∅) = 0. We shall often make free use of the
concepts and results set up so far.
The basis is the inner Daniell-Stone representation procedure in  section 7.
It assumed a function system E ⊂ [0, ∞[
with 0 ∈ E and an isotone functional I :