74.3. THE MAIN ESTIMATE 2509

And so, for ω off a set of measure zero, supt∈[0,T ] ∑i ⟨BX (t) ,ei⟩2 is bounded above. Includethis exceptional set in N.

Also for t /∈ Nω and a given ω /∈ N, letting tk→ t for tk ∈ D,

⟨BX (t) ,X (t)⟩ = ∑i⟨BX (t) ,ei⟩2 ≤ lim inf

k→∞∑

i⟨BX (tk) ,ei⟩2

= lim infk→∞

⟨BX (tk) ,X (tk)⟩ ≤ supt∈D⟨BX (t) ,X (t)⟩

and sosupt /∈Nω

⟨BX (t) ,X (t)⟩ ≤ supt∈D⟨BX (t) ,X (t)⟩ ≤ sup

t /∈Nω

⟨BX (t) ,X (t)⟩

From 74.3.15,supt /∈Nω

⟨BX (t) ,X (t)⟩=C (ω) a.e.ω

where∫

ΩC (ω)dP < ∞. In particular, supt /∈Nω

⟨BX (t) ,X (t)⟩ is bounded for a.e. ω say forω /∈N where N includes the earlier sets of measure zero. This shows that BX (t) is boundedin W ′ for t ∈ NC

ω .If v ∈V, then for ω /∈ N,

limt→s⟨BX (t) ,v⟩= ⟨BX (s) ,v⟩ , t,s

Therefore, since for such ω, ∥BX (t)∥W ′ is bounded for t /∈ Nω , the above holds for allv ∈W also. Therefore, for a.e. ω, t → BX (t,ω) is weakly continuous with values in W ′

for t /∈ Nω .Note also that∫ T

0

∫Ω

∥BX (t)∥2 dPdt ≤∫

Ω

∫ T

0∥B∥1/2 ⟨BX (t) ,X (t)⟩dtdP

≤C(||Y ||K′ , ||X ||K , ||Z||J ,∥⟨BX0,X0⟩∥L1(Ω)

)∥B∥1/2 T (74.3.16)

Eventually, it is shown that in fact, the function t→ BX (t,ω) is continuous with valuesin W ′. The above shows that BX ∈ L2 ([0,T ]×Ω,W ′).

Finally consider the claim of weak continuity of BX into W ′. From the integral equa-tion, BX is continuous into V ′. Also t → BX (t) is bounded in W ′ on NC

ω . Let s ∈ [0,T ] bearbitrary. I claim that if tn → s, tn ∈ D, it follows that BX (tn)→ BX (s) weakly in W ′. Ifnot, then there is a subsequence, still denoted as tn such that BX (tn)→ Y weakly in W ′ butY ̸= BX (s) . However, the continuity into V ′ means that for all v ∈V,

⟨Y,v⟩= limn→∞⟨BX (tn) ,v⟩= ⟨BX (s) ,v⟩

which is a contradiction since V is dense in W . This establishes the claim. Also this showsthat BX (s) is bounded in W ′.

|⟨BX (s) ,w⟩|= limn→∞|⟨BX (tn) ,w⟩| ≤ lim inf

n→∞∥BX (tn)∥W ′ ∥w∥W ≤C (ω)∥w∥W

Now a repeat of the above argument shows that s→ BX (s) is weakly continuous into W ′.