(λn)n ∈ Q. Then for arbitrary (λn)n ∈ Q, m ∈ N and every ω ∈ Ω
m∞ m m
X1(ω)(1 - Xλn) ≤ X λnXn(ω) - XλnXn(ω) ≤ X(ω)(1 - Xλn).
n=1 n=1 n=1 n=1
This implies
∞m
(*) lim Λ(P λnXn) -Λ(P λnXn) =0 for anyΛ∈ βρ-1(R).
m→∞ n=1 n=1
Let us define ∆c := {αρ ≤ c}. Then in view of Lemma 6.1 and statement .1 there exists some real c* > —ρ(0) with
ρ(Y) = max (-Λ(Y) - βρ(Λ)) = max (-EP [Y] - αρ(P))
Λ∈{βρ≤c*} p∈δc*
∞
for Y ∈ {X, P λnXn | (λn )n ∈ Q}.
n=1
The sequence (fn)n in R{βρ≤c }, defined by fn(Λ) = —Λ(Xn) — βρ(Λ), is uniformly bounded because
-2c - p(-X) ≤ — ∕βρ(Λ.) - p(-X) - ∕βρ(Λ.) ≤ fn (Λ.) ≤ ρ(Xn) ≤ ρ(Xι).
∞
Furthermore for (λn)n ∈ Q the mapping λnfn is well defined with
n=1
∞ m m∞
X λnfn(Λ) = lim (—Λ(X λnXn) — βρ(Λ) X λn) = —Λ(X λnXn) — βρ(Λ)
m→∞
n=1 n=1 n=1 n=1
for βρ (Λ) ≤ c* due to (*). Hence
∞∞ ∞ ∞
sup X λnfn (Λ) = ρ(X λnXn) = max (—Ep^X λnXn] — αp (P)) = max X λnfn(Ep∣X).
Л^вр^*} n = 1 n = 1 p∈δc* n=ι p∈δc* n = ι
The application of Simons’ lemma (cf. [20], Lemma 2) and the monotone convergence theorem leads then to
ρ(X) = sup (—EP[X] — αρ(P)) = sup limsupfn(EP∣X) ≥ inf{ sup f(Λ) ∣ f ∈ co({fn ∣ n ∈ N})},
P∈∆c. P∈∆c. n Λ∈{βρ≤c*}
where co({fn ∣ n ∈ N}) denotes the convex hull of {fn ∣ n ∈ N} in R{βρ≤c }. Thus for ε > 0 there is some convex
rr
combination f = λifni with n1 < ... < nr and ρ(X) + ε > sup f(Λ) = ρ( λifni ). In particular the
i=1 Λ∈{βρ≤c*} i=1
inequalities ρ(X) + ε > ρ(Xn ) ≥ ρ(X) hold for n ≥ nr , which implies lim ρ(Xn ) = ρ(X).
n→∞
proof of .4 ⇒ .3:
Let X+b gather all nonnegative X ∈ Xb . Drawing on Corollary 3.3 it suffices to prove that every linear form
Λ from the domain of βρ is representable by a probability measure. So let Λ belong to βρ-1 (R). Then by part
of statement .4 as well as Proposition 6.6, the linear form Λ is representable by some probability content Q on
σ(X) in the sense as introduced just before Lemma 6.5. We want to apply Proposition A.1 (cf. appendix A) to
∞ ∞ ∞ -1
verify that Q is a probability measure. Since Ω\ T Xn1([xn, ∞[) = S S (xn — (Xn ∧ Xn)) ([m1, ∞[) holds
n=1 n=1 m=1
for every pair (Xn )n , (xn)n of sequences in L ∩ X+b and ]0, ∞[ respectively, it remains to show by assumption
22
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