Fenchel-Legendre transform βρ and its counterpart αρ for the probability measures.
Lemma 6.1 Let X1, X2 ∈ X with X1 ≤ X2. Then there exists some c* ∈] — ρ(0), ∞[ such that the representation
ρ(Y) =
max
Λ∈{βρ≤c*}
(—Λ(Y) — βρ(Λ)) holds for every Y ∈ X with X1 ≤ Y ≤ X2. Moreover, for every Y ∈ X with
X1 ≤ Y ≤ X2 we have βρ(Λ) ≤ c* if ρ(Y) = -Λ(Y) — βρ (Λ).
Proof:
Let Y ∈ X with X1 ≤ Y ≤ X. By Proposition 1.1 there is some Λ ∈ β-1(R) with ρ(Y) = —Λ(Y) — βρ(Λ). Then
βρ(Λ) = —Λ(2Y) — ρ(Y) + Λ(Y) ≤ ρ(2Y) + βρ(Λ) — ρ(Y) + Λ(Y) = ρ(2Y) — 2ρ(Y) ≤ ρ(2Xχ) — 2ρ(X2).
Therefore any c > max{ρ(2X1) — 2ρ(X2), — ρ(0)} is as required. I
Lemma 6.2 Let X ∈ X with X ≤ inf Z, where E ⊆ X is assumed to be downward directed, i.e. for Z1 , Z2 ∈ E
Z∈E
there is some Z ∈ E with Z ≤ min{Z1 , Z2}. Furthermore let Λ ∈ βρ-1 (R).
Then inf Λ(Z) = Λ(X) if inf p(—λ(Z — X)) = ρ(0) for arbitrary λ > 0.
Z∈E Z∈E
Proof:
For arbitrary λ > 0 and every Z ∈ E we have βρ(Λ) ≥ —A(—λ(Z — X)) — p(—λ(Z — X)), and therefore by
assumption
0 ≤ inf Λ(Z — X) ≤ βρ(ΔL÷2p(0) .
Z∈E λ
Finally, by taking λ ↑ ∞, we obtain inf Λ(Z — X) = 0 because 0 ≤ βρ (Λ) + ρ(0) < ∞. The proof is now complete.
Z∈E
We may divide the domain of βρ into the classes {βρ ≤ c} (—p(0) = inf βρ < c < ∞). The following topological
property of these classes is crucial.
Lemma 6.3 {βρ ≤ c} is compact w.r.t. the product topology on RX for every c ∈] — ρ(0), ∞[.
Proof:
Let c ∈] — ρ(0), ∞[, and let (Λi)i∈I be a net in {βρ ≤ c} which converges to some Λ ∈ RX w.r.t. the product
topology. Obviously, Λ is a positive linear form on X which extends π. Furthermore
—Λ(X) — ρ(X) = lim(—Λi(X) — ρ(X)) ≤ limsupβρ(Λi) ≤ c for X ∈ X.
ii
Therefore {βρ ≤ c} is closed w.r.t. the product topology, and the proof may be completed by the application of
Tychonoff’s theorem because {βρ ≤ c} ⊆ X [—c — ρ(X),c + ρ(XX)]. И
X ∈ X
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