representable by a probability content if there is some probability content Q such that every X ∈ X is
integrable w.r.t. Q and Λ(X) = EQ [X].
If X is a Stonean vector lattice, then X ∧ Y = min{X, Y }, X ∨ Y = max{X, Y } ∈ X for X, Y ∈ X in particular
X+ := X ∨ 0, X- := (-X) ∨ 0 ∈ X for any X ∈ X. In this case, if all linear forms from the domain of βρ are
representable by probability contents, then ρ is concentrated on the bounded positions, and as a consequence ρ
admits a robust representation by probability measures if its restriction to the bounded positions does so.
Lemma 6.5 Let X be a Stonean vector lattice, and let every linear form Λ ∈ βρ-1 (R) be representable by some
probability content Q on σ(X). Then we can state:
.1 The sequence ρ(X+ - (X- ∧ n)) n converges to ρ(X) for every X ∈ X.
.2 The sequence ρ((X ∧ m) - Y ) m converges to ρ(X - Y ) for nonnegative X, Y ∈ X, Y being bounded.
.3 inf{∣ρ((X+ ∧ m) — (X- ∧ n)) — ρ(X)∣ ∣ m, n ∈ N} = 0 for X ∈ X, and in addition
sup (-EQ [X] - ρ(X)) = sup (-EQ [X] - ρ(X))
X∈X X∈Xb
for every probability content Q on σ(X) such that each X ∈ X is integrable w.r.t. Q .
.4 If M ⊆ αρ-1 (R) with ρ(X) = sup (—EQ [X] — αρ(Q)) for all bounded X ∈ X, then
Q∈M
ρ(X) = sup (—EQ [X] — αρ (Q)) for all X ∈ X.
Q∈M
Proof:
The most important tool of the proof is Greco’s representation theorem. The reader is kindly referred to [14]
(Theorem 2.10 with Remark 2.3).
Since any Λ ∈ βρ-1 (R) is representable by a probability content, statement .2 follows immediately from Greco’s
representation theorem and Lemma 6.4.
proof of .1:
Let X ∈ X, and let ε > 0. Then there exists some probability content Q with βρ(EQ |X) < ∞ such that the
inequality ρ(X) — ε < —EQ [X+] — βρ(EQ |X) + EQ [X-] holds. Application of Greco’s representation theorem leads
then to
ρ(X) — ε< -Eq[X+] — βρ(EQ∣X)+limEq[X- ∧ n] ≤ limρ(X+ — (X- ∧ n)) ≤ ρ(X)
nn
proof of .3:
Let Q be a probability content on σ(X) such that every X ∈ X is integrable w.r.t. Q, and let X ∈ X. Then for
ε > 0 we may choose by statement .1 and Greco’s representation theorem some n ∈ N with
∣ — Eq [X] — ρ(X) — '—Eq [X+ — (X- ∧ n)] — ρ(X+ — (X- ∧ n))´ ∣ < ɪ
17