6.2 The Predictable Range 87F 0 -measurably parameterised subspace ofRdwhereXtakes its values. This
idea was used in [S 92]. The problem is twofold. First we want to describe, in
an easy way, how subspaces depend onω∈Ω in a measurable way. Second
we have to find in some sense the smallest subspace whereXtakes its values.
The first problem can be solved in the following way. With each subspace
ofRdwe associate the orthogonal projection onto it and then we simply have
to ask that the matrices describing these projections (with respect to a fixed
orthonormal basis ofRd) depend onωin anF 0 -measurable way. There is a
more elegant way of describing subspaces using Grassmannian manifolds but
this would bring us too far from our goal.
We start with a measure theoretic result.Lemma 6.2.1.Let(Ω,F,P)be a probability space andE⊂L^0 (Ω,F,P;Rd)
a subspace closed with respect to convergence in probability. We suppose that
E satisfies the following stability property. Iff, g ∈E andA ∈F,then
f (^1) A+g (^1) Ac∈E.
Under these assumptions there exists anF-measurable mappingP 0 tak-
ing values in the orthogonal projections inRd, so thatf∈Eif and only if
P 0 f=f.
Given anyF-measurable projection-valued mappingPso thatPf=ffor
allf∈E, we have thatP 0 =P 0 P=PP 0 , meaning that the range ofP 0 is a
subspace of the range ofP.
The proof is rather a formality but requires a lot of technical verifications.
We start with a sublemma.
Sublemma 6.2.2.Under the assumptions of Lemma 6.2.1 we have, forf∈E
and a real-valuedF-measurable functionh, thathf∈E.
Proof.∑ We first prove the statement for elementary functionsh.Soleth=
n
k=1ak^1 Akwherea^1 ,...,anare real numbers andA^1 ,...,Anform anF-
measurable partition of Ω. Since the stability assumption onEimplies that
f (^1) Ak∈Efor eachk, we obviously have thathf∈Eas well.
The general case follows by approximation. Ifhis real-valued andF-
measurable we take a sequence (hn)∞n=1of elementaryF-measurable random
variables so thathn→hin probability. Clearly the closedness ofEtogether
withhnf∈Efor eachn, implieshf∈E.
Proof of Lemma 6.2.1.The construction of the projection-valued mapping
P 0 will be done through the construction of appropriate orthogonal random
vectors having maximal support. The construction is done recursively and we
first takeE 1 =E. We then look at the family ofF-measurable sets
{{f=0}|f∈E 1 }.
SinceE 1 is closed this system of sets is stable for countable unions and hence
there is an elementA 1 with maximal probabilityP[A 1 ] and with correspond-
ing functionf∈E 1 , meaning{f=0}=A 1. By the sublemma the function