B.3. THE YOUNG INTEGRAL 383
Of course there was absolutely nothing special about [0,T ] . We could have used the interval[s, t] just as well and concluded that for P a dissection of [s, t] ,∣∣∣∣∫
PY dF
∣∣∣∣≤Cpq ∥Y∥V q([0,T ]) ∥F∥p,[s,t] (2.5)
Now it is possible to show the existence of the integral. By Theorem B.2.3 there existsa sequence of piecewise linear functions {Fn} which converge to F in V p′ ([0,T ]) where p′
is chosen larger than p but still 1p′ +
1q > 1. Then letting ε > 0 be given, there exists N such
that if n≥ N,∣∣∣∣∫P
Y d (F−Fn)
∣∣∣∣= ∣∣∣∣∫P
Y dF−∫
PY dFn
∣∣∣∣≤Cp′q ∥Y∥V q ∥F−Fn∥p′,[0,T ] <ε
3
because Fn converges to F in V p′ . Also, there exists δ such that if |P| , |P ′|< δ ,∣∣∣∣∫P
Y dFN−∫
P ′Y dFN
∣∣∣∣< ε
3
Then you have, for such P , P ′,∣∣∣∣∫P
Y dF−∫
P ′Y dF
∣∣∣∣≤ ∣∣∣∣∫P
Y dF−∫
PY dFN
∣∣∣∣+
∣∣∣∣∫P
Y dFN−∫
P ′Y dFN
∣∣∣∣+ ∣∣∣∣∫P ′
Y dFN−∫
P ′Y dF
∣∣∣∣< ε
3+
ε
3+
ε
3= ε
Showing that the limit of the∫P Y dF exists as |P| → 0.
All of the above applies for any sub interval of [0,T ]. Also from 2.5, you can take alimit as |P| → 0 and conclude that
∫ ts Y dF exists and that∣∣∣∣∫ t
sY dF
∣∣∣∣≤Cpq ∥Y∥V q([0,T ]) ∥F∥p,[s,t] (2.6)
This proves the existence of the integral. Does it have the usual properties of an integral? Inparticular, if 0< a< T, is
∫ a0 Y dF+
∫ Ta Y dF =
∫ T0 Y dF? This is clearly true for the ordinary
Stieltjes integral coming from Fn. Thus from the above estimate, let {Fn} be a sequence ofpiecewise linear functions converging to F in V p ([0,T ]). Then this convergence takes placein V p ([0,a]) and V p ([a,T ]) also. Then∣∣∣∣∫ a
0Y dF−
∫ a
0Y dFn
∣∣∣∣ ≤ Cpq ∥Y∥V q([0,T ]) ∥F−Fn∥p,[0,a]∣∣∣∣∫ T
aY dF−
∫ T
aY dFn
∣∣∣∣ ≤ Cpq ∥Y∥V q([0,T ]) ∥F−Fn∥p,[a,T ]∣∣∣∣∫ T
0Y dFn−
∫ T
0Y dF
∣∣∣∣ ≤ Cpq ∥Y∥V q([0,T ]) ∥F−Fn∥p,[0,T ]
Then letting n be large enough, the right sides of the above are all less than ε. Hence, fromthe triangle inequality,∣∣∣∣∣∣
∫ a0 Y dF−
∫ a0 Y dFn +
(∫ Ta Y dF−
∫ Ta Y dFn
)+(∫ T
0 Y dFn−∫ T
0 Y dF) ∣∣∣∣∣∣< 3ε