The two stories I liked the most were Jay's Story and The Scholarship King.
Even though both of the characters in the stories were absolute opposites of each other, they
both made very good points and in the end, obtained their goals. Like how Jay kept on
Sec. 1] Bake Sets and Borel Sets 335
If X is a locally compact separable metric space, then all classes
coincide.
'FheenitiorradoptedherefortheelassofBorelsetsseemstome
to be the most useful and convenient one. It is widely used and is
standareHOHnetrie
Sec. 2] The Regularity of Baire and Borel Measures 343
It follows from Proposition 14 that the class of sets in ml with #1
nite is the same as the class of sets in 5112 with [42 nite. Conse-
-~ - when
is the same as those with respect to (,uz, 9%). Sinc
SecL2J_Ehe_Begulari1y of Baire and Borel Measures 337
d. Let X be a set with more than C elements, where C is the cardinal
ofR and let X have the discrete topology. Set 22 X x X. Then Z has the
1.20 is useful.]
12. a. For a map f: X - Y and
342 MeasurenandlopologyLChapdj
and we may take Km to be a G5 by virtue of Lemma 1. Now
uKm > #0. 2 2 ,uE 2"
> #0, 2" 2".
Set
Hm = UK.
j=m
Then Hm is a Baire set, Hm C On, C 0,l for m 2 n. Also, Hm D Hm,
and
HHmZHKm>ILOn_2n2m'
LetB-
Sec. 4] Positive Linear Functionals and Borel Measures 353
Taking the supremum over all such f gives
wsawh
1-. II I ll".
If0=01U02 with 01002=0 andfieCc(X),Osfl-sl, and
supp f,- C 0i, then the function f=f1 +f2 has supp fC 0 and
0
Sec. 3 The Existence of Invariant Measures 165
a. Show that topological equicontinuity implies CVCI'I continuity.
b. Give an example of a family 3" of maps of R to R that is eVenly
. l l . ll . . .
e. Let 5F be a family of maps of X into Y. Suppose th
350 Measure and Topology [Chap. 13
Proof: Let JC be the class of compact Gas. Then the restriction of
11 to 3% is a regular content which extends to a quasi regular Borel
11 i inner regu at or open sets
and 11 is inner regular for a bounded Baire sets,
344 Measure and Topology [Chap. 13
Before concluding this section, we remark that there are Borel
measures on compact spaces which are not regular. The construction
ofsuchanexamplea'smmewhantrieate, andwewiilnotdo'rthere.
The interested reader will nd s
Sea 21 the Regularity of Bajre and Borel Measures 34]
abounded Baire sets. We summarize this material in the following
Proposition, whose proof is left to the reader (Problem 15).
13. Proposition: Let u be a Baire measure on X. Then there is a
u
358 Measumndlopology [Qhap.13
Hence FM 3 lv1(X). But
IVI(X) S u1(X) + MIX)
=F+(D+F_(1)=HF11.
Thus HFll = lVI(X)-
Toshowthetrrriquenessofv,wenotetl=ratif,vl andvzwefeboth
nite signed Baire measures such that
dew=F(f)
for i = 1, 2 and fa C(X), t
364 LmLaLianLMeasuLes [Chap 14
g1(u) a U0, and so hl ofog[U0] C h1[0]. This implies that
f [U ] C 0, and we have shown the t0pological equicontinuity of G at
x and y. I
Weleavetheproefofthefellewmgeerelharyetothereader.
topological space X,
Sec. 5] Bounded Linear Functionals on C(X) 355
a. Show that
ankexwwnde X W=J[I$x)kx,yld]$(yl 44
= I WOU k(x, yWy) dV] d#
for all (p a CC(X) and a: e Cc(Y).
b, ShowthataaholdsforqpaCoeXyandweCo(Y).
c. If the integral in (a) is zero for all (p and 1/1 in
338 Measure and Topology [Chap. 13
For compact spaces X there is complete symmetry between inner
regularity and outer regularity: A measurable set E is outer regular if
and only if its - - .
inner regular if and only 17 1t is outer regular (and hence re
Sec. 3] The Construction of Borel Measures 345
18. Prove Proposition 14. (See Problem 17a.)
19. a. Prove Proposition 15 for the case when u is inner regular.
b. Show that the conclusion of Proposition 15 holds for measurable
sets E that are of anite
336 Measuraandlopology [ChapLS
b. What are the Baire sets in X?
c. Let X* be the one-point compactication of X. What is C(X*)?
d. What are the Baire subsets of X*?
e. Let a) be the point at innity in X*. Then cfw_to is a compact
Borel set which cont
334 Measure and Topology [Chap. 13
other hand, for each Baire subset B of X with B C F we have B =
B n F, and so B is a Baire subset of F.
T] lBlsetsis llll"ll,'E
instead of (Ba, and using the fact that F is a Borel set. I
The following lemma
340 Measine and Topology [Chap 13
Hence
0 ~ E C W ~ E,
and
140 ~ E) < e.
u 1 1 , all bounded Ba1re sets are 1n (R
Baire set is a countable unlon of bounded Ba1re sets, we see that
every aboundetLBairesetbelongs to (B. |
If we had d
Sec. 2] Topological Equijontinuity 353
is topologically equicontinuous at x. The following proposition
expresses a useful prOperty of an equicontinuous family:
1. Proposition: Let f be a family of maps from X to Y which is
topologicallyeeguieontmu
352 Measure and Topology [Chap. 13
d. Iffa CC(X) andf20, then jfdu=0ifand only lffEO on F.
[Hint: The set cfw_x: f (x) > 0 is a a-bounded Baire set.]
e. Give an example to show that F need not be a Baire set.
f. It follows from (c) that if X is co
3130 Measumandlopology [Qhag13
Let G. be an algebra of real-valued continuous funcuons on a compact
space X which separates points and contains the constants. Let (1* be the
setofsignedBaiLemeasuresonXsuchthatLyleX)slandjfdy=0foraU
fed.
aUsetheH
356 Measure and Topology [Chap. 13
on L, there are two positive linear functionals F + and F _ such that
F=F+ F_ and HF =F+(l)+F_(1).
Proof: For each nonnegat'rvefinfdene
F+(f) = SUP FM?)-
05Sf
Then F+(f)20,aITdF+(f)2F(f)- MOTCUVET
4 Invariant Measures
1Hbmb1jeTrecms Spaces
Let X be a locally compact Hausdor space. A group G of homeo-
morphisms of X onto itself is said to be transitive on X 1T, given any
two elements x, y of X, there is a homeomorphism g e G with
362 Lnyatiam Mm [Chap. 14
group G on R to consist of all linear functions g(x) = ax + b with
a > 0, then any two intervals [a, B) and [y, 6) are congruent under G.
W must give the same measurftwfj
[0,1), and [1, 2). But the rst interval is the disjoint
Sec. 2J The Regularity of aaire andJiorel Measures 339
and so
,u(0~E)<Z,u(0,~E,)<.
Thus E satises (i).
If for some n we have pEn 00, then there are compact 05s of
aLbitrar9LlaLgeJithmeasurexonlamedinEHCEHence(iiJholds
forE.prEn<ooforeachn, tlrcre'rs
Sec3]Ihernstmcon41L&oMeagnes 349
and supp (p1 C 01- Let Gi=cfw_x:(p,-(x)21/n. Then each G,- is a
compact Ga and thus in 3%. Consequently, each G, n K is in R. We
arsohave
KUanK
i=1
and G,- n K C 01- Thus
5 21m n K)
i=1
3 21701-5 2170
i=1 i=1
Taking
Sec. 3] The Construction of Borel Measures 3417
Let 0 be any open set with u*0 < co and 6 an arbitrary positive
number. Then 0 n F is an open set of nite outer measure. By pro-
perty)ofLemmal7thereisan0penset UWhUOFand
,u*U > u*(0 n F) e.
SetV_0~U.ThenV
Sec. 5| Bounded Linear Eunctionals cm C(X) 357
We always have HFII S |F+|l + |F_ II = F+(1) + F_(1). To estab-
lish the inequality in the Opposite direction, let (p be any function in
Lsuchthats<psl.Thenl21pHS Land
NF 2 F(2<P 1) = 2F(<P) - 17(1).
"Eaking1
348 MeasuLe and Topology [Chap 13
The Proposition asserts that such a set function can always be
extended to a Borel measure. A dual procedure is to start with a
suitablseffuncoromon11mfsetmompact 66s. This leads us
to the following denition.
346 Measure and Topology [Chap. 13
we generalize the procedures used in obtaining them to give us
general methods of constructing Borel measures. The rst method is
tostartwithasuitableeutermeasureandtakethemeasurablesw
with respect to this outer measure