Kenneth Kuttler

782 CHAPTER 29. MARTINGALES

Also, if a,b ∈W ′, and X ,Y ∈ L1 (Ω;W )

aE (X |S )+bE (Y |S ) = E (aX +bY |S ) . (29.3)

Proof: To begin with consider 29.3. By definition, if F ∈S ,∫F

aE (X |S )+bE (Y |S )dP = a∫

FE (X |S )dP+b

∫F

E (Y |S )dP

= a∫

FXdP+b

∫F

Y dP =∫

F(aX +bY )dP≡

∫F

E (aX +bY |S )dP

Since F is arbitrary, this shows 29.3.Let F ∈S . Then∫

FE (E (X |F ) |S )dP ≡

∫F

E (X |F )dP

≡∫

FXdP≡

∫F

E (X |S )dP

and so, by uniqueness, E (E (X |F ) |S ) = E (X |S ). This shows 29.1.To establish 29.2, note that if Z = aXF where F ∈S , and a∈W ′, by Definition 29.1.1,∫

aXF E (X |S )dP =∫

FE (aX |S )dPdP =

∫F

aXdP

=∫

aXF XdP =∫

E (aXF X |S )dP

which shows 29.2 in the case where Z is aXF ,F ∈S . It follows this also holds for simplefunctions with values in W ′. Let Z be in L∞ (Ω;W ) . By Theorem 24.2.4 there is a sequenceof simple functions {sn}, ∥sn (ω)∥ ≤ 2∥Z (ω)∥ which converges to Z and let F ∈S . Thenby what was just shown,∫

FsnE (X |S )dP =

∫F

E (snX |S )dP≡∫

FsnXdP (29.4)

Now ∥∥∥∥∫FE (snX |S )dP−

∫F

E (ZX |S )dP∥∥∥∥ =

∥∥∥∥∫F(sn−Z)XdP

∥∥∥∥≤

∫F∥(sn−Z)X∥dP

and this converges to 0 by the dominated convergence theorem. Also from Theorem 24.12.1

∥snE (X |S )∥= ∥E (snX |S )∥ ≤ E (∥snX∥|S )≤ 2E (∥ZX∥|S )

which is in L1 (Ω) . Thus one can apply the dominated convergence theorem to the left sideof 29.4 and use what was just shown to pass to a limit in 29.4 and obtain∫

FZE (X |S )dP =

∫F

ZXdP≡∫

FE (ZX |S )dP.

Since this holds for every F ∈S , this shows 29.2. ■The next major result is a generalization of Jensen’s inequality whose proof depends

on the following lemma about convex functions. It pertains to the case where the functionshave values in R.

782 CHAPTER 29. MARTINGALESAlso, if a,b € W', and X,Y € L'(Q;W)aE (X|.7)+bE (Y|.A%) = E (aX +bY|-7). (29.3)Proof: To begin with consider 29.3. By definition, if F € .7,[oe (X|7) +bE(Y|.A)dP=a | E(X|A)aP +b | E(Y|.Y)aPF F F=a] xap+b | vap= | (aX +bY)dP = | B(axX +bY|.Y) APF F F FSince F is arbitrary, this shows 29.3.Let F € .Y. ThenE(E(X|F)|P)dP E(X|F)aPI I[ xap= | E(x\r)aPand so, by uniqueness, E (E (X|.F)|.7) = E (X|.%). This shows 29.1.To establish 29.2, note that if Z = a2 where F € .Y, anda € W’, by Definition 29.1.1,/ a2rE (X|.P)dP | E (aX|.f) dPdP = | aXdPF F= / aXeXdP = / E(a&%peX|.S)aPwhich shows 29.2 in the case where Z isa 2p,F € .%. It follows this also holds for simplefunctions with values in W’. Let Z be in L® (Q;W). By Theorem 24.2.4 there is a sequenceof simple functions {s,}, ||s,(@)|| <2||Z(@)|| which converges to Z and let F € .Y. Thenby what was just shown,| sa (X|.P)dP = | E (spX|.)dP = | snXdP (29.4)F F JF| [ o-zoxar|[.iilon-Z)x\laPand this converges to 0 by the dominated convergence theorem. Also from Theorem 24.12.1Now| [eoxizyar— | eexir)ar|IAIlnE (X|-P)|| = ||E (snX|%)|| < E (|lsnX |] |) < 2 (||ZX] |)which is in L! (Q). Thus one can apply the dominated convergence theorem to the left sideof 29.4 and use what was just shown to pass to a limit in 29.4 and obtain[zewnyar = | zxap= | e(zx| rar,Since this holds for every F € .Y, this shows 29.2. HiThe next major result is a generalization of Jensen’s inequality whose proof dependson the following lemma about convex functions. It pertains to the case where the functionshave values in R.