63.5. THE QUADRATIC VARIATION AND STOCHASTIC INTEGRATION 2143

Then letting

Xτ ln,m (t) =

∣∣∣∣∣∣∣∣∫ t

0( fn− fm)dMτ l

∣∣∣∣∣∣∣∣U,

Xn,m (t) =

∣∣∣∣∣∣∣∣∫ t

0( fn− fm)dM

∣∣∣∣∣∣∣∣U

=

∣∣∣∣∣∣∣∣∫ t

0fndM−

∫ t

0fmdM

∣∣∣∣∣∣∣∣U

It follows Xτ ln,m is a continuous nonnegative submartingale and from Theorem 62.9.4 just

listed,

P([

Xτ l∗n,m (T )> λ

])≤ 1

λ2

∫Ω

Xτ ln,m (T )2 dP

≤ 1

λ2

∫Ω

∫ T

0| fn− fm|2 d [Mτ l ]dP

≤ 1

λ2

∫Ω

∫ T

0| fn− fm|2 d [M]dP

Letting l→ ∞,

P([

X∗n,m (T )> λ])≤ 1

λ2

∫Ω

∫ T

0| fn− fm|2 d [M]dP

Therefore, there exists a subsequence, still denoted by { fn} such that

P([

X∗n,n+1 (T )> 2−n])< 2−n

Then by the Borel Cantelli lemma, the ω in infinitely many of the sets[X∗n,n+1 (T )> 2−n]

has measure 0. Denoting this exceptional set as N, it follows that for ω /∈ N, there existsn(ω) such that for n > n(ω) ,

supt∈[0,T ]

∣∣∣∣∣∣∣∣∫ t

0fndM−

∫ t

0fn+1dM

∣∣∣∣∣∣∣∣≤ 2−n

and this implies uniform convergence of{∫ t

0 fndM}

. Letting

G(t) = limn→∞

∫ t

0fndM,

for ω /∈N and G(t) = 0 for ω ∈N, it follows that for each t, the continuous adapted processG(t) equals

∫ t0 f dM a.e. Thus

{∫ t0 f dM

}has a continuous version.