The Mathematics of Arbitrage

9.8 Appendix: Some Measure Theoretical Lemmas 203

with the compact (metrisable) space [0,∞]. A sequence (xn)n≥ 1 of elements
of [0,∞] is a Cauchy sequence in [0,∞] if and only if for eachα>0thereis
n 0 so that for alln, m≥n 0 we have|xn−xm|≤αor min(xn,xm)≥α−^1.
From the properties ofuit also follows that forα>0thereisβ>0, so that
|x−y|>αand min(x, y)≤α−^1 , impliesu

(x+y

2

)

^12 (u(x)+u(y)) +β.
We can now easily proceed with the proof of the lemma. By the observation
on the topology of [0,∞] we have to show that limn,m→∞P[|gn−gm|>α
and min(gn,gm)≤α−^1 ]=0.
For givenα>0wetakeβas above and we obtain

E

[

u

(

gn+gm

2

)]

≥

1

2

E[u(gn)] +

1

2

E[u(gm)]

+βP[|gn+gm|>αand min(gn,gm)<α−^1 ].

By constructionE

[

u

(gn+gm

2

)]

≤sn, but by concavity ofuwe have

E

[

u

(

gn+gm

2

)]

≥

1

2

(

E[u(gn)] +E[u(gm)]

)

.

From this it follows

βP

[

|gn−gm|>αand min(gn,gm)<α−^1

]

≤E

[

u

(

gn+gm

2

)]

−

1

2

(

E[u(gn)] +E[u(gm)]

)

.

The choice of the sequence (gn)n≥ 1 implies that the right hand side tends
to 0. We therefore proved that (gn)n≥ 1 is a Cauchy sequence in probability
and hence there is a functiong:Ω→[0,∞]sothatgnconverges togin
probability. If one wants a sequence converging almost surely one can pass to
a subsequence.
If conv{fn;n≤ 1 }is bounded inL^0 then for eachε>0thereisN so
thatP[h>N]<εfor allh∈conv{fn;n≥ 1 }. In particular this implies
thatP[gn>N]<εand henceP[g>N]≤ε. The functiongso obtained is
therefore finite almost surely.
IfP[fn >α]>δ>0foreachnand fixedα>0, we obtain that
E[u(gn)]≥δu(α)>0. Sincegntends togwe findu(gn)→u(g)andby
the bounded convergence theorem we obtainE[u(g)]≥δu(α)>0 and there-
foreP[g>0]>0.

Remark 9.8.2.If (fn)n≥ 1 is a sequence of [0,∞]-valued measurable functions
then the same conclusion can be obtained. The proof is the same up to minor
changes in the notation. The reader can convince himself that there is almost
no gain in generality.

Remark 9.8.3.If (fn)n≥ 1 is a sequence ofR-valued measurable functions such
that conv{fn−;n≥ 1 }is bounded inL^0 , then there aregn∈conv{fn;n≥ 1 }so
thatgnconverges almost surely to a ]−∞,+∞]-valued measurable functiong.