852 CHAPTER 31. OPTIONAL SAMPLING THEOREMS

2. There exists M (∞) ∈ L1 (Ω) such that limt→∞ ∥M (∞)−M (t)∥L1(Ω) = 0.

In this case, M (t) = E (M (∞) |Ft) and convergence also takes place pointwise.

Proof: Suppose the equi-integrable condition. Then there exists λ large enough thatfor all t, ∫

[|M(t)|≥λ ]|M (t)|dt < 1.

It follows that for all t,∫Ω

|M (t)|dP =∫[|M(t)|≥λ ]

|M (t)|dP+∫[|M(t)|<λ ]

|M (t)|dP

≤ 1+λ .

Since the martingale is bounded in L1, by Theorem 31.5.3 there exists M (∞) ∈ L1 (Ω)such that limt→∞ M (t)(ω) = M (∞)(ω) pointwise a.e. By the assumption {M (t)} areequi-integrable, it follows from Proposition 10.9.6 these functions are uniformly integrable.Then by the Vitali convergence theorem, Theorem 10.9.7, if tn→ ∞, then

∥M (tn)−M (∞)∥L1(Ω)→ 0

Next suppose there is a function M (∞) to which M (t) converges in L1 (Ω) . Then for tfixed and A ∈Ft , then as s→ ∞,s > t∫

AM (t)dP =

∫A

E (M (s) |Ft)dP≡∫

AM (s)dP

→∫

AM (∞)dP =

∫A

E (M (∞) |Ft)

which shows E (M (∞) |Ft) = M (t) a.e. since A ∈Ft is arbitrary. By Theorem 24.12.1,∫[|M(t)|≥λ ]

|M (t)|dP =∫[|M(t)|≥λ ]

|E (M (∞) |Ft)|dP

≤∫[|M(t)|≥λ ]

E (|M (∞)| |Ft)dP

=∫[|M(t)|≥λ ]

|M (∞)|dP (31.7)

Now from this,

λP([|M (t)| ≥ λ ]) ≤∫[|M(t)|≥λ ]

|M (t)|dP≤∫

Ω

|E (M (∞) |Ft)|dP

≤∫

Ω

E (|M (∞)| |Ft)dP =∫

Ω

|M (∞)|dP

and soP([|M (t)| ≥ λ ])≤ C

λ

From 31.7, this shows {M (t)} is equi-integrable hence uniformly integrable because thisis true of the single function |M (∞)|. ■