On infinite-dimensional generalizations of Lyapunov’s
theorem
Erik Balder, University of Utrecht
1. LT seen as a bang-bang principle
(Ω, Σ, µ) : finite measure space
Recall Lyapunov’s theorem (LT): if µ non-atomic then
Z
{ f : E ∈ Σ} is convex and compact set
E
for every f ∈ L1(Ω; Rd).
Related versions of LT: bang-bang principle (BBP) and
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Aumann’s identity. Many applications: control theory,
economics, game theory, etc.
Bad news: direct infinite-dimensional extension of LT fails.
Good news: infinite-dimensional extension of BBP works.
Recall close relationship between LT and BBP:
Let Φ := {φ ∈ L∞(Ω) : φ ∈ [0, 1] a.e.}; then
ext Φ = {φ ∈ Φ : φ ∈ {0, 1} a.e.} = {1E : E ∈ Σ}.
BBP says: if µ non-atomic then
Z
Z
Z
{ f φ : φ ∈ Φ} = { f φ : φ ∈ ext Φ} = { f : E ∈ Σ}.
Ω
Ω
E
2
for every f ∈ L1(Ω; Rd).
Fact:
LT ⇒ BBP and BBP + Alaoglu-Bourbaki ⇒ LT.
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2. Kingman-Robertson’s BBP
Notation: Σ+ := {E ∈ Σ : µ(E) > 0}.
For E ∈ Σ+ can obviously embed L∞(E) ⊂ L∞(Ω).
Central idea of K-R: localized non-injectivity replaces
non-atomicity.
KR-Theorem. Let {fα : α ∈ A} ⊂ L1(Ω) be such
that for every E ∈ Σ+
Z
φ 7→ ( φfα)α∈A
E
from L∞(E) into RA is non-injective. Then following
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BBP holds:
Z
Z
{( φfα)α : φ ∈ Φ} = {( φfα)α : φ ∈ ext Φ}.
Ω
Ω
Proof of K-R similar to Lindenstrauss’ proof of LT by use
of extreme point arguments.
If A finite and µ non-atomic, then KR’s localized noninjectivity condition holds. So KR-theorem implies LT.
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3. Generalization of K-R’s BBP
S : metrizable Lusin space
Ŝ: Hilbert cube compactification of S; identify S ⊂ Ŝ;
then S ∈ B(Ŝ)
M (Ŝ): set of all bounded signed Borel measures on Ŝ
M+≤1(Ŝ) ⊂ M (Ŝ): set of all subprobability measures on
Ŝ.
Notation: L := L∞(Ω; M (Ŝ)) set of all bounded signed
transition measures from Ω into Ŝ.
Fact: L is dual space of Bochner space L1(Ω; C(Ŝ)), so
can be equipped with weak star topology.
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Let Γ : Ω → 2S be multifunction.
Assumptions for Γ:
graph Γ := {(ω, s) ∈ Ω × S : s ∈ Γ(ω)} ∈ Σ ⊗ B(S),
Γ(ω) ⊂ S is nonempty and compact a.e.
Let LΓ := {κ ∈ L : |κ(ω)|tv (Ŝ\Γ(ω)) = 0 a.e.}.
Let ΦΓ := {κ ∈ LΓ : κ(ω) ∈ M+≤1(Ŝ) ⊂ M (Ŝ) a.e.}.
Then ext ΦΓ = {κ ∈ LΓ : κ(ω) ∈ ext M+≤1(Ŝ) a.e.}
Consequence: ext ΦΓ consists of those κ ∈ LΓ for which
for a.e. ω the subprobability measure κ(ω) is either null
measure or Dirac probability measure.
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Theorem 1. Let Y be a TVS and let Λ : LΓ → Y be
a linear operator that is weak star continuous on ΦΓ and
such that for every E ∈ Σ+ and κ ∈ LΓ
φ 7→ Λ(φκ)
from L∞(E) into Y is non-injective. Then
Λ(ΦΓ) = Λ(ext ΦΓ).
The KR-theorem follows as special case from Theorem 1:
take S = Ŝ a singleton; say S = {s0}. Then ΦΓ can be
identified with the earlier set Φ, because every
ω 7→ κ(ω)({s0})
forms a unique element of Φ and vice versa. Also, in this
specialization LΓ coincides with L∞(Ω). We use Y := RA
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and
Z
κ(ω)({s0})fα(ω)µ(dω))α.
Λ(κ) := (
Ω
Finally, non-injectivity of mappings à la K-R easily implies non-injectivity as needed in Theorem 1.
Theorem can be stated more abstractly. Just as proof of
K-R, it essentially follows Lindenstrauss’ proof of LT by
use of extreme point arguments. Important inspiration
from paper of Rustichini-Yannelis (1991).
Sufficient condition for localized noninjectivity:
Hamel-dimension of Y < Hamel-dimension of L∞(E)
for every E ∈ Σ+.
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4. Applications to AI and FLID
Observe: can rewrite original BBP (section 1) as Aumann
identity (AI):
Z
Z
co {f, 0} = {f, 0},
Ω
Ω
d
where for any multifunction Γ : Ω → 2R
Z
Z
Γ := { ψ : ψ ∈ L1(Ω; Rd), ψ ∈ Γ a.e.}.
Ω
Ω
Fact: original AI ⇔ original BBP.
Recall: original AI (1965) states
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Z
Z
co Γ =
Ω
Γ,
Ω
d
for every “integrably bounded below” Γ : Ω → 2R with
closed values and measurable graph.
Now obtain a version of AI, namely
Z
Z
cl co Γ =
Γ,
Ω
Ω
for Γ : Ω → 2X , where X is a separable Banach space.
Assumptions for Γ:
graph Γ := {(ω, x) ∈ Ω × X : x ∈ Γ(ω)} ∈ Σ ⊗ B(X),
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Γ(ω) ⊂ X is nonempty and σ(X, X ∗)-compact a.e.
Γ is integrably bounded,
i.e., there exists ψ ∈ L1(Ω) with x ∈ Γ(ω) ⇒ kxk ≤
ψ(ω).
Theorem 2. If Γ is as above and if
Hamel-dimension of X < Hamel-dimension of L∞(E)
for every E ∈ Σ+, then
Z
Z
cl co Γ =
Γ,
Ω
Ω
where
Z
Ω
Z
Γ := { ψ : f ∈ L1(Ω; X), f ∈ Γ a.e.}
Ω
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etc.
Result follows from application of Theorem 1 to S :=
(X, σ(X, X ∗)).
Sketch proof of nontrivial inclusion: fix f ∈ L1(Ω; X)
with f (ω) ∈ co Γ(ω) a.e.
R
R
Desired to show Ω f ∈ Ω Γ.
For each ω there is a probability measure κω on X with
Z
f (ω) = bar κω :=
x κω (dx).
Γ(ω)
Can choose ω 7→ κω measurably; then κ ∈ ΦΓ.
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By Theorem 1 and suitable formulation of weak star continuous Λ: κ̄ ∈ ext ΦΓ exists with
Z
Z
Z
bar κ̄ =
bar κ [=
f ].
Ω
and
Z
Ω
Z
κ̄(·)(X) =
Ω
Ω
κ(·)(X) [= µ(Ω)].
Ω
Second property guarantees that a.e. κ̄(ω) is probability
on Γ(ω) (not just sub-probability).
But then κ̄(ω) is Dirac probability measure on Γ(ω) a.e.
Hence f¯ := bar κ̄ is measurable a.e. selector of Γ and
desired result follows from first property.
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Another application: Fatou’s lemma in infinite dimensions:
Theorem 4. Let {fn}n be a sequence in L1(Ω; X) such
that fn(ω) ∈ Γ(ω)
a.e. for every n and such that a :=
R
σ(X, X ∗) − limn Ω fn exists. If dim X < dim L∞(E) for
every E ∈ Σ+,Rthen there exists a function f∗ ∈ L1(Ω; X)
such that a = Ω f∗ and for a.e. ω in Ω
f∗(ω) is a σ(X, X ∗)-limit point of {fn(ω)}n.
In other words:
Z
Z
Γ⊂
LS-
LS-Γ,
Ω
Ω
where LS refers to σ(X, X ∗)-sequential limes superior in
Kuratowski’s sense.
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