As an application of Lemma 6.3 we may show the following useful technical argument.
Lemma 6.4 Let E ⊆ X consist of nonnegative positions, and let E be upward directed, i.e. for Z1 , Z2 ∈ E there is
some Z ∈ E with Z ≥ max{Z1 , Z2}. Furthermore let X := sup Z ∈ X, and let Y ∈ X be nonnegative and bounded.
Z∈E
Then inf ρ(Z - Y) = ρ(X - Y) holds if inf (Λ(X) - Λ(Z)) = 0 for every Λ ∈ βρ-1(R).
Z∈E Z∈E
Proof:
^ ∙ ι*∙.
Due to Lemma 6.1 there exists some real c > —ρ(0) with ρ(X — Y) = sup (—Λ(X — Y) — βρ(Λ)) for all
Λ∈{βρ≤c*}
tz- — τn I I Γ tz"i mi ι ι
X ∈ E ∪ {X }. Then we may conclude
0 ≤ inf ρ(Z - Y) - ρ(X - Y) ≤ inf sup FZ (Λ),
Z∈E Z∈E Λ∈{βρ≤c* }
where FZ : {βρ ≤ c*} → R, Λ → Λ(X) — Λ(Ζ), for Z ∈ E.
In the view of Lemma 6.3 ({βρ ≤ c*},τ) is a compact Hausdorff space, where τ denotes the relative topology
of the product topology on RX to {βρ ≤ c*}. Since E
directed family of real-valued mappings, i.e. for Z1 , Z2
Furthermore all functions from M are continuous w.r.t.
is upward directed, M := {FZ | Z ∈ E} is a downward
∈ E there exists some Z ∈ E with FZ ≤ min{FZ1 , FZ1 }.
τ, and inf FZ (Λ) = 0 for Λ ∈ {βρ ≤ c*} by assumption.
Z∈E
Therefore the application of the general Dini lemma (cf. [12], Theorem 3.7) leads to inf sup FZ (Λ) = 0,
Z∈E Λ∈{βρ≤c*}
which completes the proof.
In the next step we want to look for conditions which allow to reduce investigations to bounded positions. For this
purpose we have to recall some concepts from integration theory, adapted to our setting. If Q denotes a probability
content on the σ—algebra σ(X), i.e. a finitely additive nonnegative set function with Q(Ω) = 1, then we shall call
a a(X) —measurable mapping X with positive and negative part X+ and X- integrable w.r.t. Q if
∞
Z Q({X+
0
≥ x}) dx,
Q({X- ≥ x}) dx < ∞.
The terminology stems from the fact that Q may be extended via the so-called asymmetric Choquet integral
EQ defined by ([6], Chapter 5, p. 87)
∞
∞
Q({X- ≥ x}) dx.
0
EQ[X] := Q({X+ ≥ x}) dx —
0
It is a positive linear form on the space of all Q —integrable mappings (cf. [6], Proposition 5.1, Theorem 6.3,
Corollary 6.4), and hence the restriction to the bounded ones is even continuous w.r.t. to the sup norm. Therefore
the restriction of Eq to the bounded a(X)—measurable mappings is just the respective integral defined in functional
analysis (e.g. [7], Appendix A.5). Using the introduced notions, a real linear form Λ on X is defined to be
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