Chapter 61

Probability In Infinite Dimensions61.1 Conditional Expectation In Banach Spaces

Let (Ω,F ,P) be a probability space and let X ∈ L1 (Ω;R). Also let G ⊆F where G isalso a σ algebra. Then the usual conditional expectation is defined by∫

AXdP =

∫A

E (X |G )dP

where E (X |G ) is G measurable and A ∈ G is arbitrary. Recall this is an application of theRadon Nikodym theorem. Also recall E (X |G ) is unique up to a set of measure zero.

I want to do something like this here. Denote by L1 (Ω;E,G ) those functions inL1 (Ω;E) which are measurable with respect to G .

Theorem 61.1.1 Let E be a separable Banach space and let X ∈ L1 (Ω;E,F ) where X ismeasurable with respect to F and let G be a σ algebra which is contained in F . Thenthere exists a unique Z ∈ L1 (Ω;E,G ) such that for all A ∈ G ,∫

AXdP =

∫A

ZdP

Denoting this Z as E (X |G ) , it follows

∥E (X |G )∥ ≤ E (∥X∥ |G ) .

Proof: First consider uniqueness. Suppose Z′ is another in L1 (Ω;E,G ) which works.

Consider a dense subset of E {an}∞

n=1. Then the balls{

B(

an,∥an∥

4

)}∞

n=1must cover E \

{0}. Here is why. If y ̸= 0, pick an ∈ B(

y, ||y||5

).

y an

0

Then ||an|| ≥ 4 ||y||/5 and so ||an− y||< ||y||/5. Thus

y ∈ B(an, ||y||/5)⊆ B(

an,||an||

4

)Now suppose Z is G measurable and ∫

AZdP = 0

1985