The Mathematics of Arbitrage

9.8 Appendix: Some Measure Theoretical Lemmas 205

Lemma 9.8.6.Let(gk) 1 ≤k≤nbe non-negative functions defined on the prob-
ability space (Ω,F,P). Suppose that there are positive numbers(ak) 1 ≤k≤nas
well asδ> 0 so that for everyk:P[gk≥ak]≥δ> 0 .Ifg=

∑n
j=1gjthen
for all 0 <η< 1 we haveP[g≥η(

∑n

j=1aj)δ]≥

δ(1−η)

1 −ηδ.

Proof.LetA={g≥(

∑n

j=1aj)δη}. Clearly

E[g (^1) Ac]≤
( n
∑
j=1
aj

)

δηP[Ac]=

( n

∑

j=1

aj

)

δη(1−P[A]).

On the other hand

E[g (^1) Ac]=
( n
∑
j=1
E[gj (^1) Ac]

)

≥

( n

∑

j=1

ajP

[

Ac∩{gj≥aj}

]

)

≥

( n

∑

j=1

aj

(

P[gj≥aj]−P[A]

)

)

≥

( n

∑

j=1

aj

)

δ−

( n

∑

j=1

aj

)

P[A].

Both inequalities imply

( n

∑

j=1

aj

)

P[A](1−δη)≥

( n

∑

j=1

aj

)

δ(1−η).

We may of course suppose that

∑n
j=1aj>0 and this yields the desired
resultP[A]≥δ 1 (1−−δηη).

Corollary 9.8.7.If(gj) 1 ≤j≤nare non-negative functions defined on the prob-
ability space (Ω,F,P) and if forj=1,...,nwe haveP[gj≥a]≥bwhere
a, b > 0 , then forg=

∑n

j=1gjwe haveP

[

g≥nab 2

]

≥ 2 b.

Acknowledgement

The authors want to thank P. Artzner, M.Emery, P. M ̈ ́ uller and Ch. Stricker
for fruitful discussions on this paper. Part of this research was supported by
the European Community Stimulation Plan for Economic Science contract No
SPES-CT91-0089.