c ∈]ρ(0), ∞[. So (Ep∣X)i∈ι is a net in {βρ ≤ c}, which in turn is compact w.r.t. to product topology on RX by
Lemma 6.3. Therefore there exist a subnet (Pi(k))k∈K and some Λ ∈ {βρ ≤ c} with lim EPi(k) [X] = Λ(X) for
k ()
every X ∈ X. Then (Pi(k))k∈K converges to some P ∈ ∆c due to (*). This finishes the prove of statement .2.
proof of .3:
Drawing on Lemma 6.5 with Proposition 6.6 and the translation invariance of ρ it remains to show
ρ(X) = sup (-EP [X] - αρ (P)) for all X ∈ X+b.
P∈M1(S)
For this purpose let X ∈ X+b , and let ε > 0. Then by (1), (2) and definition of E we may find an isotone sequence
(Yn)n in L+b with X ≤ sup Yn ∈ E and ρ(X) < inf ρ(Yn) + ε. In view of Lemma 6.1 and (*) there is some
nn
c* ∈] — ρ(0), ∞[ with ρ(Yn) = sup (—EP[Yn] — αρ(P)) for any n. Furthermore Fn(P) := — EP[Yn] — αρ(P) defines
P∈∆c.
an antitone sequence of mappings Fn := ∆c* → R which are upper semicontinuous w.r.t. the relative topology of
TL (see Lemmata 7.2, 6.5 again) such that Fn ∖ F, defined by F (P) := —Ep[Y ] — αρ(P). Due to .2 we may apply
the generalized Dini lemma (cf. [12], Theorem 3.7) and obtain
ρ(X) — ε < infρ(Yn) = sup (—EP [Y] — αρ(P)) ≤ sup (—EP[X] — αρ(P)) ≤ ρ(X),
n P∈∆c, P∈∆c,
which completes the proof. H
Proof of Remark 3.4:
Obviously, σ(X) = B(Ω) ® B(T), and any closed subset A of Ω × T w.r.t. the metrizable topology tω × TT may be
∞
described by A = T Xn-1 ([xn , ∞[) for some sequence (Xn)n of nonnegative continuous mappings and a sequence
n=1
(xn)n of positive real numbers.
Now let X ∈ L. Then X(∙,t) is B(Ω)—measurable for t ∈ T and X(ω, ∙) is continuous w.r.t. tt for ω ∈ Ω, which
implies X ∈ X because TT is separably metrizable (cf. e.g. [3], Lemma III-14). Therefore L ⊆ X, and σ(X) is
generated by S. Then the statement of Remark 3.4 follows immediately from Theorem 3.1. H
8 Proof of Theorem 4.1
Obviously, .2 ⇒ .3, and statement .2 implies statement .4 if the indicator mappings 1A (A ∈ S) belong to X. So
in view of Corollary 3.3 and Proposition 1.1 it remains to prove the implications .1 ⇒ .2 and .4 ⇒ .3.
proof of .1 ⇒ .2:
∞
Let Q consist of all sequences (λn)n in [0, 1] with λn = 1. Aditionally, let (Xn)n be any isotone sequence in
n=1
∞
X which converges pointwise to some X ∈ X. Then by assumption λnXn is a well defined member of X for
n=1
21