276 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory
Remark 13.6.3.The previous theorem shows thatGis a num ́eraire invariant
space provided we only accept num ́eraire changes induced by maximal admis-
sible contingent claims.
13.7 The Closure ofG∞and Related Problems
In this section we will study the contingent claims ofKmaxthat are in the
closure ofG∞. The characterisation is done using either uniform convergence
over the setMeor using the setMba. Before we start the program, we first
recall some notions from integration theory with respect to finitely additive
measures; we refer to Dunford-Schwartz [DS 58] for details.
Letμbe a finitely additive measure that is inba(Ω,F,P). A measurable
functionf(we continue to identify functions that are equalP-a.s.), defined
on Ω is calledμ-measurable if for eachε>0 there is a bounded measurable
functiongsuch thatμ{ω||f(ω)−g(ω)|>ε}<ε. The reader can check
that sinceFis aσ-algebra, this definition coincides with [DS 58, Definition
10]. We say that aμ-measurable functionfisμ-integrable if and only if there
is sequence (gn)n≥ 1 of bounded measurable functions such thatgnconverges
inμ-measure tofandsuchthatEμ[|gn−gm|] tends to zero ifn, mtend to
∞. In this case one definesEμ[f] = limn→∞Eμ[gn]astheμ-integralEμ[f]
off.Incasefis bounded from below theμ-integrability offimplies via the
dominated convergence theorem, valid also for finitely additive measures, that
E[f−f∧n] tends to zero asntends to∞. Contingent claimsgofG∞are
μ-integrable for allμ∈Mbaand moreover we trivially haveEμ[g]=0since
EQ[g] = 0 for allQ∈Me.
Proposition 13.7.1.Suppose that S is a locally bounded semi-martingale
that satisfies the (NFLVR) property. Iff∈Kmaxandμ∈Mba,thenf is
μ-integrable andEμ[f]≤ 0 .Alsoμ[f ≥n]≤^4 ‖nf‖, a uniform bound over
μ∈Mba. In particular for eachμ∈Mbaand eachf∈Kmaxwe find that
f∧ntends tofinμ-measure andEμ[f∧n]tends toEμ[f]asntends to
infinity.
Proof.We only have to prove the statement for contingent claimsfthat
are 1-admissible and maximal. So suppose thatfis such a contingent claim.
By the optional stopping theorem, or by the maximal inequality for super-
martingales, we find that for allQ∈Me,wehavethatQ[f ≥n]≤^1 n.
The setMeisσ(ba, L∞)-dense inMba(see Remark 9.5.10), hence we obtain
thatμ[f ≥n]≤^1 n for alln.Sinceμ[f−f∧n>0]≤μ[f ≥n]≤^1 n,
the measurability follows for functionsfthat are 1-admissible and maximal.
The general case follows by splittingfasf=g−hwhere eachgandhare
‖f‖-admissible and by the fact that{|f|>n}⊂{|g|>n 2 }∪{|h|>n 2 }.
To see that forμ∈Mba, the integralEμ[f] exists and is negative, let us
first observe that for allnand allQ∈Mewe have thatEQ[f∧n]≤0. This