http://www.geocities.jp/narunarunarunaru/study/etc0.jpg
にアップしておりましたが大変失礼いたしました。

問題9は

「Let C([a,.b]) denote the vector space of continuous functions on the
closed and bounded interval [a,b].
Suppose we are given a Borel measure μ on this interval,with μ([a,b])
<∞. Then
f→L(f)=∫_a^b f(x) dμ(x)
is a linear functional on C([a,b]),with L positive in the sense that L
(f)≧0 if f≧0.
Prove that,conversely,for any linear functional L on  C([a,b]) that is
positive in the above sense,there is a unique finite Borel measure μ
so that L(f)=∫_a^b fdμ for f∈C([a,b]).
[Hint: Suppose a=0 and u≧0. Define F(u) by F(u)=lim_{ε→0} L(f_ε),where
f_ε(x)=1(for 0≦x≦u), 0(for u+ε≦x)
and f_ε is linear between u and u+ε.(See Figure 3.) Then F is
increasing and right-continuous,and L(f) can be written as ∫_a^b f(x)dF
(x) via Therem3.5.]
The result also holds if [a,b] is replaced by a closed infinite
interval; we then assume that  L is defined on the continuous
functions of bounded support,and obtain that the resulting μ is finite
on all bounded intervals.
A generalization is given in Problem 5.」

でございます。

下記normalizedの定義とTheorem3.5の説明です。
「The Stieltjes integral was introoudeced to provide a generalization
of the Riemann integral ∫_a^b f(x)dx,whrere the incerments dx were
replaced by the increments dF(x) for a given increasing function F on
[a,b].We wish to pursue this idea from the general point of view taken
in this chapter.The quastion that is then raised is that of
characterizing the measures on R that arise in this way,and in
priticular measures defined  on the Borel sets on the real line.
 To have a unique correspondence between measures and increasing
functions as we shall have below,we need first to normalize these
functions appropriately.Recall that an incrasing funciton F can have
at most a countable number of discontuities.If x_0 is such a
discontinuity,
then
lim[x<x_0,x→x_0」F(x)=F(x_0^-) and lim[x>x_0,x→x_0」F(x)=F(x_0^+)
both exist,while F(x_0^-)<F(x_0^+) and F(x_0) is some value between F
(x_0^-) and F(x_0^+).We shall now modify F at x_0,if necessary,by
setting F(x_0)=F(x_0^+),and we do this for  every point of
discontinuity.The function F so obtained is now still increasing,yet
right-continuous at every poingt,and we say such functions are
normalized. The main result is then as folllows.
Theorem3.5
Let F be an increasing function on R that is normalized. Then there is
a unique measure μ (also denoted by dF) on the Borel sets B on R such
that μ((a,b])=F(b)-F(a) if a<b. Cnonversely, if μis a measure on B
that is finite on bounded intervals,then F defined  by F(x)=μ
((0,x]),x>0,F(0)=0 and,F(x)=-μ((-x,0]),x<0,is increasing and
normalized.」