The Mathematics of Arbitrage

202 9 Fundamental Theorem of Asset Pricing

EP

[

lim inf

j→∞

( (^1) Ajmin(f 0 ,1))

]

≤EP

[

lim inf

j→∞

(^1) Ajmin((Lj·S)∞,1)

]

≤lim inf

j→∞

EP

[

(^1) Ajmin((Lj·S)∞,1)

]

≤lim inf

j→∞

EP

[

(^1) Ajmin((Lj·S)m 0 ,1)

]

+

∑

m>m 0

EP

[

min((Lj·S)m,1)−min((Lj·S)m− 1 ,1)

]

≤−

βm 0

m 0

+

∑

m>m 0

2 m+1m^2 αm

≤−

βm 0

2 m 0

.

This is clearly a contradiction tof 0 ≥0.

9.8 Appendix: Some MeasureTheoreticalLemmas ..............

In this appendix we prove two lemmas we used at several places. We assume
that, especially regarding the second lemma, the results are known, but we
could not find a reference. We therefore give full proofs and we also add some
remarks that are of independent interest but are not used elsewhere in this
paper. The first lemma was already proved in [S 92, Lemma 3.5]. We give
a similar but simpler proof.

Lemma 9.8.1.Let(fn)n≥ 1 be a sequence of[0,∞[-valued measurable func-
tions on a probability space(Ω,F,P).Thereisgn∈conv{fn,fn+1,...},such
that(gn)n≥ 1 converges almost surely to a[0,∞]-valued functiong.
Ifconv{fn;n≥ 1 }is bounded inL^0 ,thengis finite almost surely. If there
areα> 0 andδ> 0 such that for alln:P[fn>α]>δ,thenP[g>0]> 0.

Proof.Letu:R+∪{+∞}→[0,1] be defined asu(x)=1−e−x.Economists
may seeuas a utility function but there is no need to. Definesnas

sn=sup{E[u(g)]|g∈conv{fn,fn+1,...}}

and choosegn∈conv{fn,fn+1,...}so that

E[u(g)]≥sn−

1

n

.

Clearlysndecreases tos 0 ≥0 and limn→∞E[u(gn)] =s 0 .Weshallshowthat
the sequence (gn)n≥ 1 converges in probability to a functiong. We will work