90 6 The Dalang-Morton-Willinger Theorem
inL^0 (Ω,F,P;K), i.e., a sequence ofK-valued functions depending onω∈Ω
in a measurable way. Here (K,d) is equipped with its Borelσ-algebra.
We would like to find a subsequence (nk)∞k=1so that (fnk(ω))∞k=1converges
for eachω(or at least for almost eachω). If Ω is finite, this is clearly possible.
But considering a sequence (fn)∞n=1of independent Bernoulli variables i.e.,
P[fn =1]=P[fn =−1] =^12 , we see that in general this wish is asking
for too much. For each subsequence (nk)∞k=1the sequence (fnk)∞k=1still forms
an independent sequence of Bernoulli variables and therefore diverges almost
surely. On the other hand, for each fixedω∈Ω, we may of course extract a
subsequence (nk(ω))∞k=1such that (fnk(ω))∞k=1converges. The crux is, that
the choice of the subsequence depends onω, as we just have seen. The idea of
the subsequent notion of a measurably parameterised subsequence is that this
subsequence (nk(ω))∞k=1may be chosen to dependmeasurablyonω∈Ω. This
nice and simple idea was observed by H.-J. Engelbert and H. v. Weizs ̈acker
and was successfully applied by Y.M. Kabanov and Ch. Stricker ([KS 01]) in
the present context.
Definition 6.3.1.AnN-valued,F-measurable function is called a random
time. A strictly increasing sequence(τk)∞k=1of random times is called amea-
surably parameterised subsequenceor simply ameasurable subsequence.
Before stating the actual result we first mention the following lemma on
random times.
Lemma 6.3.2.Let(fn)∞n=1be a sequence ofF-measurable functionsfn:Ω→
K.Letτ:Ω→{ 1 , 2 , 3 ,...}be anF-measurable random time, theng(ω)=
fτ(ω)(ω)isF-measurable.
Proof.LetBbe a Borel set inK.Then
g−^1 (B)=⋃∞
n=1({τ=n}∩{fn∈B})∈F. Proposition 6.3.3.For a sequence(fn)∞n=1∈L^0 (Ω,F,P;K)we may find a
measurably parameterisedsubsequence(τk)∞k=1such that(fτk)∞k=1converges
for allω∈Ω.
Proof.It suffices to check that the procedure of finding a convergent subse-
quence in the proof of the Bolzano-Weierstrass Theorem may be done in a
measurable way.
Forn≥1, letAn 1 ,...,AnNnbe finite open coverings ofKby sets of diam-
eter less thann−^1. For each fixedω∈Ω we define inductively a sequence
(Ik(ω))∞k=1 of infinite subsets of N.Fork = 1 find the smallest number
1 ≤j 1 ≤N 1 such that (fn(ω))∞n=1lies infinitely often inA^1 j 1 , and define
I^1 (ω)={n∈N|fn(ω)∈A^1 j 1 }.