This dates from about 1936. Basically, you can do ∫_{0}^{T}Y dF if Y is continuous and F is of
bounded variation or the other way around. This is the old Stieltjes integral. However, this
integral has to do with Y ∈ V^{q} and F ∈ V^{p} and of course, these functions are not necessarily
of bounded variation although they are continuous. First, here is a simple lemma which is
used a little later.
Lemma 10.3.1Let F be piecewise linear on a dissection P and let Y be continuous.Then
∫ t
t → Y dF
0
is continuous.
Proof:Say P is given by
{t0,t1,⋅⋅⋅,tn}
where 0 = t_{0}<
⋅⋅⋅
< t_{n} = T. Let
∫ t
G (t) ≡ YdF
0
Then on
[0,t1]
,
∫ t ∫ t
G (t) = Y (s) F-(t1)−-F-(0)ds = F (t1)−-F (0) Y (s)ds
0 t1 − 0 t1 − 0 0
which is clearly continuous. Then on
[t1,t2]
you have
∫
t F (t2)−-F-(t1)
G (t) = G (t1)+ t1 Y (s) t2 − t1 ds
which is again continuous. Continuing this way shows the desired conclusion. ■
Definition 10.3.2Let P be a dissection of
[0,T ]
and let Y,F be continuous.Then
∫ ∑
Y dF ≡ Y (ti)(F (ti+1) − F (ti))
P P
where P =
{t0,⋅⋅⋅,tn}
. This is like a Riemann Stieltjes sum except that you don’t have abounded variation integrator function.∫_{0}^{T}Y dF is said to exist if there is I ∈ ℝ suchthat
|∫ |
lim || Y dF − I||= 0
|P|→0 | P |
meaning that for every ε > 0 there exists δ > 0 such that whenever
|P |
< δ, it follows that
||∫ YdF − I||
P
< ε. It suffices to show that for every ε there exists δ such that if
|P|
,
|P′|
< δ,then
||∫ ∫ ||
|| YdF − ′ Y dF|| < ε
P P
This last condition says that the set of all these∫_{P}for
|P|
sufficiently small has smalldiameter.
The following theorem is from Young.
Theorem 10.3.3Let 1 ≤ p,q,
1p
+
1q
> 1. Also suppose that F ∈ V^{p}
([0,T ])
andY ∈ V^{q}
([0,T])
. Then for all sub interval
[a,b]
of
[0,T]
there exists I_{[a,b]
}such that
|∫ |
lim || Y dF − I||= 0
|P|→0 | P |
exists where here P⊆
[a,b]
. Thus we are considering dissections of
[a,b]
. Also there existestimates of the form
|∫ |
|| t ||
| s YdF | ≤ Cpq∥Y ∥V q([0,T])∥F ∥p,[s,t] (10.3)