2090 CHAPTER 62. STOCHASTIC PROCESSES
By Proposition 62.7.8, Fσ∧τ ⊆Fσ , and so since A ∈Fσ was arbitrary,
E (X (τ) |Fσ )≥ X (σ ∧ τ) a.e.
Note that a function defined on a countable ordered set such as the integers or equallyspaced points is right continuous.
Here is an interesting lemma.
Lemma 62.7.16 Suppose E (|Xn|) < ∞ for all n, Xn is Fn measurable, Fn+1 ⊆ Fn forall n ∈ N, and there exist X∞ F∞ measurable such that F∞ ⊆ Fn for all n and X0 F0measurable such that F0 ⊇Fn for all n such that for all n ∈ {0,1, · · ·} ,
E (Xn|Fn+1)≥ Xn+1, E (Xn|F∞)≥ X∞.
Then {Xn : n ∈ N} is uniformly integrable.
Proof:E (Xn+1)≤ E (E (Xn|Fn+1)) = E (Xn)
Therefore, the sequence E (Xn) is a decreasing sequence bounded below by E (X∞) so it hasa limit. Let k be large enough that∣∣∣E (Xk)− lim
m→∞E (Xm)
∣∣∣< ε (62.7.26)
and suppose n > k. Then if λ > 0,∫[|Xn|≥λ ]
|Xn|dP =∫[Xn≥λ ]
XndP+∫[Xn≤−λ ]
(−Xn)dP
=∫[Xn≥λ ]
XndP+∫
Ω
(−Xn)dP−∫[−Xn<λ ]
(−Xn)dP
=∫[Xn≥λ ]
XndP−∫
Ω
XndP+∫[−Xn<λ ]
XndP
From 62.7.26,≤∫[Xn≥λ ]
XndP−∫
Ω
XkdP+ ε +∫[−Xn<λ ]
XndP
By assumption,E (Xk|Fn)≥ Xn
and so
≤∫[Xn≥λ ]
E (Xk|Fn)dP−∫
Ω
XkdP+ ε +∫[−Xn<λ ]
E (Xk|Fn)dP
=∫[Xn≥λ ]
XkdP−∫
Ω
XkdP+ ε +∫[−Xn<λ ]
XkdP
=∫[Xn≥λ ]
XkdP−∫
Ω
XkdP+ ε +∫[Xn>−λ ]
XkdP
=∫[Xn≥λ ]
XkdP+
(∫Ω
(−Xk)dP−∫[Xn>−λ ]
(−Xk)dP)+ ε
=∫[Xn≥λ ]
XkdP+∫[Xn≤−λ ]
(−Xk)dP+ ε =∫[|Xn|≥λ ]
|Xk|dP+ ε