2402 CHAPTER 70. MEASURABILITY WITHOUT UNIQUENESS
Proof: Let Pm denote the projection onto the closed ball B(0,9m). Then from the abovelemma, there exists a product measurable solution um to the integral equation
um (t,ω)−u0 (ω)+∫ t
0N(s,Pmum(s,ω),Pmum(s−h,ω),Pmwm (s,ω) ,ω)ds
=∫ t
0f(s,ω)ds.
Define a stopping time
τm(ω)≡ inf{
t ∈ [0,T ] : |um (t,ω)|2 + |wm (t,ω)|2 > 2m},
where inf /0≡ T . Localizing with the stopping time,
uτmm (t,ω)−u0 (ω)+
∫ t
0X[0,τm]N(s,uτm
m (s,ω),uτmm (s−h,ω),wτm
m (s,ω) ,ω)ds
=∫ t
0X[0,τm]f(s,ω)ds.
Note how the stopping time allowed the elimination of the projection map in the equation.Then we get
12|uτm
m (t,ω)|2− 12|u0(ω)|2
+∫ t
0
(X[0,τm]N(s,uτm
m (s,ω),uτmm (s−h,ω),wτm
m (s,ω) ,ω) ,uτmm (s,ω)
)ds
=∫ t
0X[0,τm] (f(s,ω) ,uτm
m (s,ω))ds.
From the estimate,
12|uτm
m (t,ω)|2− 12|u0(ω)|2 ≤
∫ t
0
(µ
(|uτm
m (s,ω)|2 + |uτmm (s−h,ω)|2 + |wτm
m (s,ω)|2)
+C (s,ω)+12|f(s,ω)|2
)ds+
12
∫ t
0|uτm
m (s,ω)|2 ds.
Note that|u0|2 h+
∫ t
0|uτn
n (s)|2 ds≥∫ t
0|uτn
n (s−h,ω)|2 ds
and ∫ t
0|wτn
n (s,ω)|2 ds =∫ t
0
∣∣∣∣w0 +∫ s
0X[0,τn]un (r)dr
∣∣∣∣2 ds
=∫ t
0
∣∣∣∣w0 +∫ s
0X[0,τn]u
τnn (r)dr
∣∣∣∣2 ds
≤C (w0 (ω))+CT∫ t
0|uτn
n |2 ds