Kenneth Kuttler

1946 CHAPTER 60. CONDITIONAL, MARTINGALES

Let F ∈S . Then∫F

E (E (X |F ) |S )dP ≡∫

FE (X |F )dP

≡∫

FXdP≡

∫F

E (X |S )dP

and so, by uniqueness, E (E (X |F ) |S ) = E (X |S ). This shows 60.1.1.To establish 60.1.2, note that if Z = XF where F ∈S ,∫

XF E (X |S )dP =∫

XF XdP =∫

E (XF X |S )dP

which shows 60.1.2 in the case where Z is the indicator function of a set in S . It followsthis also holds for simple functions. Let {sn} be a sequence of simple functions whichconverges uniformly to Z and let F ∈S . Then by what was just shown,∫

FsnE (X |S )dP =

∫F

snXdP =∫

FE (snX |S )dP

Now ∣∣∣∣∫FE (snX |S )dP−

∫F

E (ZX |S )dP∣∣∣∣

≤∫

F|snX−ZX |dP =

∫F|sn−Z| |X |dP

which converges to 0 by the dominated convergence theorem. Then passing to the limitusing the dominated convergence theorem, yields∫

FZE (X |S )dP =

∫F

ZXdP≡∫

FE (ZX |S )dP.

Since this holds for every F ∈S , this shows 60.1.2.The next major result is a generalization of Jensen’s inequality whose proof depends on

the following lemma about convex functions.

Lemma 60.1.3 Let φ be a convex real valued function defined on an interval I. Then foreach x ∈ I, there exists ax such that for all t ∈ I,

φ (t)≥ ax (t− x)+φ (x) .

Also φ is continuous on I.

Proof: Let x ∈ I and let t > x. Then by convexity of φ ,

φ (x+λ (t− x))−φ (x)λ (t− x)

≤ φ (x)(1−λ )+λφ (t)−φ (x)λ (t− x)

=φ (t)−φ (x)

t− x.

1946 CHAPTER 60. CONDITIONAL, MARTINGALESLet F € Y. Then[EER|P)|Z)aP - [e@\P)aP[ xaps | exinyarF Fand so, by uniqueness, E (E (X|.F) |.7) = E (X|.%). This shows 60.1.1.To establish 60.1.2, note that if Z = 2p where F €.Y,‘/[me (X|.Y)dP = | vexaP= [el ax\y)arwhich shows 60.1.2 in the case where Z is the indicator function of a set in .”. It followsthis also holds for simple functions. Let {s,} be a sequence of simple functions whichconverges uniformly to Z and let F € .7. Then by what was just shown,[nb xin )aP = | sxaP = | B(x) dPF F FNow| [eluxinyar— fe ex)r\ar]F F< J imx —zx\ap = | |S» —Z||X|dPF Fwhich converges to 0 by the dominated convergence theorem. Then passing to the limitusing the dominated convergence theorem, yields[zex\\ar= | zxap= | E(zx\7)aP.Since this holds for every F € -Y, this shows 60.1.2.The next major result is a generalization of Jensen’s inequality whose proof depends onthe following lemma about convex functions.Lemma 60.1.3 Let @ be a convex real valued function defined on an interval I. Then foreach x € I, there exists a, such that for allt € I,@ (t) > ax (t—x) +9 (x).Also @ is continuous on I.Proof: Let x € J and let t > x. Then by convexity of ¢,P(x +A (t= x) — 0) — OO) I-A) $49 (1) — 9)A(t —x) ~ A (t—x)_ 9) (x)fax