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Unformatted text preview: 378 D Borel and Radon Measures on the Real Line D.8 The Riesz Representation Theorem for Positive
Linear Functionals on CAR) ' In this section we will discuss one version of the Riesz Representation Theo—
rem, which shows the equivalence between Radon measures and positive linear
functionals on COOK). This version is only concerned with positive measures
and positive functionals, while in Section D.10 we will see a Riesz Represen—
tation Theorem for complex Radon measures and bounded functionals. In this section we deal both with measures and functionals. Typically, we
will let n. denote a functional and u a measure. Notation D.71. In keeping with the notations introduced in Appendix C
(see Notation 0.36), we write (f, ,u) to denote the action of a linear functional
n: (IGOR) —r (C on a vector f E COOK). Further, (f, n) is a sesquilinear form,
linear in 3“ but antilinear in ,u. Each Radon measure V on R induces an associated linear functional ,u on
CAR) by the formula (fun) 2 ffdv, feCcUR). (DJ) Note that, by deﬁnition, 1/ is a positive measure here. In order to ensure that
(, ) is a sesquilinear form, whenever we extend our consideration to complex
measures we will need to replacedy by dz? in equation (D.7). This example immediately raises several questions, which we will address in
this section. First, is the functional ,u deﬁned in equation (D.7) continuous on
Cc (1R)? Of course, continuity is not even deﬁned until we specify the topology
on CAR), and, as it turns out, there is more than one “natural” choice. Second, once we specify the topology on CAR), does every continuous
linear functional on CCOR) have the form given in equation (D37)? In other
words, can we characterize the dual space of CAR)? This question also requires
some reﬁnement, since we have speciﬁed that Radon measures are positive
measures, whereas if we let 1/ be a complex measure then we can still deﬁne a
functional ,u by equation (DJ). To address these questions, we next discuss two particular topologies on one). D.8.1 Topologies on CAR) Since we wish to study the continuity of linear functionals on GC (R), we must
specify a topology or a convergence criterion on Cc (R). There are two natural
choices. (a) The unifom (or L°° mom) topology. CAR) is a normed space with respect
to the topology induced by the uniform, or L°°, norm. A linear functional .__u , _ mums... D8 The Riesz Representation Theorem for Positive Linear Functionals on 030R) 379 u on CC (IR) is continuous with respect to the uniform topology if and only
if it is bounded with respect to the If” norm. That is, u is continuous if
and only if there exists a constant C > 0 such that “fun“ 5 C llfllooa all f E COUR)‘ Since CAR) is a dense subspace of the Banach space 000R), such a ,u has
a unique extension to a bounded linear functional on all of CD (R), which
we also refer to as ,u (see Exercise C26). (b) The inductive limit topology. For each compact K g R, deﬁne
C(K) : {f E (IGOR) : supp(f) Q K}. Each C(K) is a Banach space with respect to'the I’m—norm. Further, as
a set,
CAR) '= U {C(K) : K g R, K compact}. We can deﬁnea topology on CCOR) by declaring that a function whose
domain is Cch) is continuous if for each compact K its restriction to
C(K) is continuous with respect to the Lm—norm on C(K). In particular, a
linear functional n: CAR) + (C is continuous with respect to this topology
if and only if for each compact K the restriction HICUO: C(K) *4 (C is
continuous. Since C(K) is a normed space, this happens if and only if
each plow) is bounded with respect to the norm on C(K), which means
that for each compact K there exists a constant OK > 0 such that WWI S CKllflloo: all f€C(K) (133) However, unlike boundedness with respect to the uniform topology, where
there is a single constant C that determines the boundedness, the con—
stants OK in equation (DB) can depend on the compact set K. In technical
language, this topology corresponds to the inductive limit of the topolo
gies (C(K),   ”00) with K compact, and hence we will refer to it as the
inductive limit topology on CCUR). We refer to [ConQO] for details on the
inductive limit of topologies. This type of topology is also discussed in
Section E5. The following deﬁnition of the convergence criterion on CAR) corresponds
to convergence with respect to each'of these two topologies. Deﬁnition D372. Let {fn}neN be a sequence of functions in 060R). (a) We say that fn converges to f uniformly, or in I'mnorm, if M f — fnoo —>
D. In this case, we write 3",, —> f unifomnly. (b) We say that fn converges to f in (IGOR) if there exists a compact set K such that supp(fn) g K for all n, and lif — fn¢o —> 0. In this case, we
write fn —> f in CAR). I
F
i
F
r
l 380 D Borel and Radon Measures on the Real Line In particular,
fn —> f in CAR) => fn —) f uniformly . (DB) However, the converse implication does not hold in general, so these are two
distinct topologies on COOK). Equation (D.9) implies that the uniform topol—
ogy on CAR) is weaker than the inductive limit topology. In this section we are focusing on‘Radon measures (which by deﬁnition
are positive but possibly unbounded) and corresponding positive linear func—
tionals on Co (R). For these results it is the inductive limit topology 011 GAR)
that will be important. In contrast, in Section D.10 we will consider complex
Radon measures (which are necessarily bounded) and corresponding linear
functionals on 041R), and there it will be the ISMtopology on 06(R) that will
be important. D.8.2 Positive Linear Functionals on CC(R) The next exercise shows that every Radon measure induces a linear functional
on 066R) that is continuous with respect to the inductive limit topology on
Cc (R). Further, this functional is positive in the following sense. 7 Deﬁnition D.73. A functional ,u: CCGR) ——> (C is positive if (fHu) 2 U for all
f 6 CAR) with f 2 0. Exercise D.74. Let V be a Radon measure on R. Deﬁne ,n: OAR) —» (C by (fuu) = frat», f e can). (a) Show that p, is a positive linear functional on CAR). (1)) Show that MICLK): C(K) —> R is continuous for every compact K g R,
i.e.,
V compact K <_Z 1R, EICK > 0 such that f€C(K) => lifnuHSCKllflloo Thus, those positive linear functionals on CCOR) that are induced from
Radon measures are continuous with respect to the inductive limit topology
on CAR). Next we will show directly every positive linear functional on CAR)
is continuous with respect to the inductive limit topology on 0.; (1R). (p.10) Theorem D.75. If n: CCGR) —> (C is a positive linear functional on COOK),
then ,u is continuous on CCGR) with respect to the inductive limit topology.
Speciﬁcally, MOW): C(K) —+ C is continuous for each compact K Q 1R. Proof. Given a compact set K, by Urysohn’s Lemma (Theorem 1.60 or The
orem AJOQ), we can ﬁnd 6K 6 CAR) such that 9K 2 0 and 9K 2 1 on K.
Suppose ﬁrst that f E C(K) is realvalued. Then D.8 The Riesz Representation Theorem for Positive Linear Functionals on Ca (R) f($)1 = f($)9K($) S lfoo9K($)
for all a: E R. Therefore “1'”ng :l: f 2 0, and so 0 S (llfiloogxifaﬂ) = flwi9K,M)i(f:M)
Consequently, Hf: #ll 5 (610 M) llflloo
Second, given an arbitrary f E C(K), we have l(f,#) Sr HREULHH + (1m(fliﬂ) S 2<8K1#)llfllw
Hence the result follows with OK = 2 (GK, n). III Althou we wi not prov , the Hi Repre ation T orem c —
pletes e char erizatio positive ' ear fun ' nals on c R): Ev pos—
i ' e iinea unctional CAR) ' ' duced m a R measur UR) C Theorem D.76 (Riesz Representation Theorem I). If n: Cc H
is a positive linear functional, then there crisis a unique positive Radon mea
sure u on R such that (ﬁn) = ] ray, 1" 6 CAR)
_Moreouer, if U Q R is open, then V(U) = sup{(f,,u) :f E CCURLO —<_ f S 1,Supp(f) Q U}:
and if K g IR is compact then MK) = 111mm) = f e 0.303),)“ 2 Xx} Thus, Radon measures and positive linear functionals on C’c (R) are equiva lent. Therefore, we often use the same symbol to represent a Radon measure 1/
and the positive functional f H ( f, 1/) = f f dv that it induces. Additional Problems D.25. This problem will show that the locally ﬁnite positive measures on N
(which by Problem D24 are precisely the Radon measures on N) are in 11
correspondence with the positive linear functionals on cm). (a) Give the convergence criterion corresponding to the inductive limit
topology on (300. (b) Show that if v is a positive locally ﬁnite measure on N, then (f, v) z
E ﬂit) 1/{k} deﬁnes a positive linear functional on Coo that is continuous with
respect to the inductive limit topology on Coo (c) Show that if u is a positive linear functional on 000 then there exists
a unique sequence of nonnegative scalars u) : (wakeN such that (ﬁn) =
Z f(k) wk for f E 000 Show there is a unique locally ﬁnite positive measure
1/ on N such that wk = 1/{k} for every k. 381 188 4 Distributions though it seems obvious, is it true that n is zero 0 supp(,u)? By en and n is zero on U}. , then, since supp( f) is com appen, so eed the following lemma. Lemma 4.50. Let U1, . . . , UN be open subsets oflR, and let K Q UILJ  UUN
be compact. Then them exist 9;, 6 03° (R) with supp(8k) g U}, such that Zg=1élk=1 onK. Pmof. Exercise: For each It : 1, . . . ,N, construct an open set V3, with compact
closure that satisﬁes 7;, Q ch, and such that K g V1 U    U VN. By the 0"” Urysohn Lemma (Theorem 1.60), for each R: = 1, . . .,N we
can ﬁnd a function (pk E C§°(R) such that 0 S (pg, 3 1, (pk = 1 on W, and
supp(cpk) Q Uh. We can also ﬁnd a function 1b 6 C§°(R) such that 0 S d) S 1,
1/} = 1 on K, and suppWJ) g V1 U    LJ VN. The function CECE) = (I — Wm» ‘l' 290143)
k:1 is then inﬁnity differentiable and everywhere nonzero. Since 1b = 1 on K, it
follows that the functions em) = 9:23), k=1,...,N satisfy 21:1 6;, = 1 on K. El rcise 4.51. Show that if ,u. E ’D’GR), then ,u is zero on R . rpm). As a co .auence of Exercise 4.51, supp(,u.) is the Ma lest closed set such
that ,u is zero on r? upp(p,). In particular, it f0 w s that supp(,u) = 0 if and
only if ,u = 0. Here are some of the basic r  nert‘ =; of the support of a distribution. (b) If g E L110c(]R) .. c n9 is the distribution that is . : ermined by Q, then
suppmg) = : pp(g) (see Notation 1.20 for the meaning a the support of
a funct' (c) If . E D’UR) then supp(ﬂ) = —supp(,u) and supp(Ta,u) I suppm)
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