Kenneth Kuttler

748 CHAPTER 27. ANALYTICAL CONSIDERATIONS

Now let g (y) ≡ E (X|y1, · · · ,yk) be a Borel representative of∫Rn xdλX|y It follows

ω → g (Y (ω)) = E (X|Y 1 (ω) , · · · ,Y k (ω)) is σ (Y 1, · · · ,Y k) measurable because bydefinition ω→Y (ω) is σ (Y 1, · · · ,Y k) measurable and a Borel measurable function com-posed with a measurable one is still measurable. It follows that for all E Borel in Rp,∫

Y −1(E)XdP =

∫E

E (X|y1, · · · ,yk)dλY

=∫Y −1(E)

E (X|Y 1 (ω) , · · · ,Y k (ω))dP

so Z (ω) = E (X|Y 1 (ω) , · · · ,Y k (ω)) works because a generic set of σ (Y 1, · · · ,Y k) isY −1 (E) for E a Borel set in Rp. If both Z,Z1 work, then for all

F ∈ σ (Y 1, · · · ,Y k) ,∫F(Z−Z1)dP = 0

Since F is arbitrary, some routine computations show Z =Z1 a.e. ■

Observation 27.4.5 Note that a.e.

E (X|Y 1 (ω) , · · · ,Y k (ω)) = E (X|σ (Y 1, · · · ,Y k))

where the one on the left is the expected value of X given values of Y j (ω). This onecorresponds to the sort of thing we say in words. The one on the right is an abstractconcept which is usually obtained using the Radon Nikodym theorem and its description isgiven in the lemma. This lemma shows that its meaning is really to take the expected valueofX given values for the Y k.

27.5 Characteristic Functions for MeasuresRecall the characteristic function for a random variable having values in Rp. I will givea review of this to begin with. Then the concept will be generalized to random variables(vectors) which have values in a real separable Banach space.

Definition 27.5.1 LetX be a random variable. The characteristic function is

φX (t)≡ E(eit·X)≡ ∫

Ω

eit·X(ω)dP =∫Rp

eit·xdλX

the last equation holding by Proposition 26.1.12 on Page 717.

Recall the following fundamental lemma and definition, Lemma 13.2.4 on Page 379.

Definition 27.5.2 For T ∈ G ∗, define FT,F−1T ∈ G ∗ by

FT (φ)≡ T (Fφ) , F−1T (φ)≡ T(F−1

φ)

Lemma 27.5.3 F and F−1 are both one to one, onto, and are inverses of each other.

The main result on characteristic functions is the following in Theorem 27.1.4 on Page735 which is stated here for convenience.

748 CHAPTER 27. ANALYTICAL CONSIDERATIONSNow let g(y) = E(X|y,,---,y,) be a Borel representative of fen dA x\y It followso> g(Y (@)) =E(X|Y\(@),---,Y¥,(@)) is o(Y1,---,¥,) measurable because bydefinition @ > Y (@) is o(Y1,--- , Y,) measurable and a Borel measurable function com-posed with a measurable one is still measurable. It follows that for all E Borel in R?,[sg XP = [LE Xl sereY-!(E) E=[ E(X|Y1(@),---,¥4(@))aPY~!(E)so Z(@) =E(X|Y1(@),:--,¥,%(@)) works because a generic set of o(Y1,---,Y x) isY—!(E) for E a Borel set in R”. If both Z, Z, work, then for allFeo(Yi,:::,Yx),[ (Z—Z)dP =0Since F is arbitrary, some routine computations show Z = Z, a.c.Observation 27.4.5 Note that a.e.E(X|¥1(@),---,¥x(@)) =E(X|o(¥Y1,--- ,¥x))where the one on the left is the expected value of X given values of Y ;(@). This onecorresponds to the sort of thing we say in words. The one on the right is an abstractconcept which is usually obtained using the Radon Nikodym theorem and its description isgiven in the lemma. This lemma shows that its meaning is really to take the expected valueof X given values for the Y x.27.5 Characteristic Functions for MeasuresRecall the characteristic function for a random variable having values in R?. I will givea review of this to begin with. Then the concept will be generalized to random variables(vectors) which have values in a real separable Banach space.Definition 27.5.1 Let X be a random variable. The characteristic function isOx (t)=E(e**) = | et X() gp = , eb) xQ RPthe last equation holding by Proposition 26.1.12 on Page 717.Recall the following fundamental lemma and definition, Lemma 13.2.4 on Page 379.Definition 27.5.2 for T « G*, define FT,F-'T € Y* byFT (o)=T (FO), F'T(9) =T(F'9)Lemma 27.5.3 F and F~' are both one to one, onto, and are inverses of each other.The main result on characteristic functions is the following in Theorem 27.1.4 on Page735 which is stated here for convenience.