2. Prove that Proposition 3.1 holds when X is any separable metric space, not necessarily locally compact.
Hint. Use the existence of a countable basis of open sets (Exercise 1 on page 10).
3. A particular case of the Vitali-Caratheodory Theorem. Let p be a posi- tive Radon measure on X. Prove that for everyà-integrable and bounded function f from X to R and for all c > 0, there exists an upper semi- continuous function u and a lower semicontinuous function v such that u < f < v and f (v - u) dp < e. (We say that u is upper semicontinuous if -u is lower semicontinuous.)
Hint. Go over the proof of Daniell's Theorem (page 59) and use the result in Exercise 3 on page 64.
4. Let it be a positive Radon measure on X and take f E LR (p). Prove that there exist p-integrable and lower semicontinuous functions f+ and f_
with values in [0, +oo), such that f = f+ - f- à-almost everywhere. (As in the case of real-valued functions (Exercise 3 on page 64), a function g with values in [-oo, +oo] is called lower semicontinuous if the set {g>a} is open for allaER.)
Hint. Show that there exists a sequence (con) in CR(R) that converges to f in LR (p) and à-almost everywhere and such that l` (con - cpn+i I) <
2-n for all n E N. Then set f+ = coo + En a(Vn+1 - Wn)+ and f_ _ W0 + En 0(Wn+l - fin)
5. Regularity of Radon measures. (This is a sequel to Exercise 3 on page 64.) Let p be a positive Radon measure on X.
a. Prove that, for every Borel set A of X,
p(A) = inf{JhdP : h is lower semicontinuous and h > 1A } . b. Let A be a Borel set in X such that à(A) is finite.
i. Take e > 0. Let h be a lower semicontinuous function such that h > 1A and f h dp < à(A) + e, and set
U= Ix EX:h(x)> à(A)+2e u(A)+e
Prove that A C U and that à(U) < à(A) + 2e.
78 2. Locally Compact Spaces and Radon Measures ii. Deduce that
à(A) = inf{à(U) : U is open and U D A}.
iii. Check that this is still true if p(A) = oo (this is obvious). A measure p satisfying this equality for all Borel sets A is called
outer regular.
c. Let U be an open subset of X. Prove that
à(U) = sup{à(K) : K is compact and K C U}.
Hint. U is a-compact.
d. Let A be a Borel set of finite measure à(A).
i. Let e > 0. Justify the existence of-
- an open set U in X containing A and such that à(U) <p(A)+e;
- an open set V in X containing U\A and such that u(V) < 2e;
- a compact set K in X contained in U and such that u(K) >
à(U) - e.
Finally, set C = K \ V. Prove that C C A and that à(C) >
p(A) - 3e.
ii. Deduce that
à(A) = sup{à(K) : K is compact and K C A}.
iii. Generalize to the case of an arbitrary Borel set A. A measure p satisfying this equality for all Borel sets A is called inner regular.
Hint. By exhausting X with a sequence of compact sets, prove that A is the union of an increasing sequence of Borel sets of finite measure.
e. i. Prove that for every Borel set A of X and all e > 0 there exists an open set U in X such that A C U and à(U \ A) < e.
ii. Prove that for every Borel set A of X and all e > 0 there exists an open set U and a closed set F in X such that F C A C U and u(U\F) <e.
Hint. Apply the preceding result to A and X \ A.
6. Lusin's Theorem. Let m be a positive Radon measure on X.
a. Let f be a Borel function on X with values in 10, 11. Prove that, for any open set 0 of finite measure and any e > 0, there exists a compact K C 0 such that m(O \ K) < e and the restriction f etc is continuous on K.
Hint. Use Proposition 3.5, Exercise 15 on page 155 and the fact that m is inner regular (see Exercise 5d).
b. Extend the preceding result to all Borel functions f from X to K.
Hint. First reduce to the case where f takes values in R', then consider 1 = f /(1 + f ).
c. Deduce that every Borel function f from X to K satisfies this prop- erty:
(L) For every e > 0, there exists an open set w in X such that m(w) < c and the restriction of f to X \ w is continuous.
Hint. Consider an increasing sequence (On)nEN of relatively com- pact open sets that covers X. For each n, there exists a compact Kn C On for which m(On \ K1t) < e2-n-' and f IK. is continuous.
Now set w = Un(O1 \ Kn). Prove that (X \ w) n O C Kn for every n; then conclude the proof.
d. Show that a function f from X to K satisfies Property L if and only if there exists a Borel function that equals f rn-almost everywhere.
Hint. To prove sufficiently, use the fact that m is outer regular (Ex- ercise 5b).
7. a. Let it be a positive Radon measure on X, with support F. Let f E CC(X) be such that f (x) = 0 for all x E F. Prove that f f dp = 0.
b. Let A = {an}n<N be a finite subset of X and p a positive Radon measure on X. Prove that the support of a equals A if and only if p is a linear combination of Dirac measures ba with positive coefficients.
c. Let A = {an} be a countable subset of X. For f E Cc (X) write p(f) = E 2-nf(an).
nEN
Prove that p is a positive Radon measure on X whose support is the closure of A.
8. a. Let F be a closed subset of X. Prove that F is the support of a continuous function f from X to R if and only if F coincides with the closure of F.
b. Let p be a positive Radon measure on X. W e denote b y 2 (p) the space of locally p-integrable functions on X, by which we mean Borel functions t : X -+ K such that 1K>Ji E 2'(p) for any compact K of X. (For example, every continuous function on X is locally p- integrable.) Fix a Eli E Yi(p) taking nonnegative values. For f E CC(X), write
i(f) =J4'fd.
Prove that v is a positive Radon measure. Prove that Supp v c {ip 0} n Supp p, with equality if 0 is continuous.
c. For f E CC(R2), write
v(f) = jf(xx)dz.
80 2. Locally Compact Spaces and Radon Measures
Prove that v is a positive Radon measure on R2 and determine its support.
Is there a continuous function ip on R2 such that v(f) = JR2 f (x, y) dx dy for all f E Cc(R2)?
9. a. Let m be a positive linear form on CR (X ). Show that there exists a compact K in X such that any f E CR (X) that vanishes on K satisfies m(f) = 0.
Hint. Exhaust X by a sequence (Kn) of compact sets. Show that, if there is no K as stated, there exists a sequence (fn) of elements of C+(X) such that, for each n E N, the function fn vanishes on Kn and m(fn) > 0. Then consider f = EnEN fn/m(fn)
b. Let 0+(X) be the set of positive Radon measures with compact support. To every p E 931 (X), associate the positive linear form mà on CR (X) defined by
Jfd
m,,(f)=à f or f ECR(X).
Prove that the map p ,-> m,, is a bijection between 9X+(X) and the set of positive linear forms on CR (X).
Hint. See the proof of Proposition 3.6 (page 71) for inspiration.
LO. Vague convergence. We say that a sequence (pn)nEN of positive Radon measures on X converges vaguely to p E 911+(X) if
An(f) -+à(f) for allf E Cc(X).
a. An example. Let (an)nEN be a sequence in X with no cluster point.
Prove that the sequence (San )nEN converges vaguely to 0.
b. Another example. Suppose X = (0, 1). Prove that the sequence (An) defined by
n-1 An = n E 6k/n
k=1
converges vaguely to Lebesgue measure on (0, 1).
c. Let (àn) be a sequence in 9X+ (X) such that, for all f E CC (X), the sequence (An(f )) converges. Prove that the sequence (pn) is vaguely convergent.
d. Let p be a positive Radon measure and A a relatively compact Borel set whose boundary has p-measure zero. Prove that, if (àn)nEN is a sequence in 9A+ (X) that converges vaguely to p, then
niimpàn(A) = à(A).
Hint. Show the existence of an increasing sequence in CC (X) that converges pointwise to the characteristic function of A, and of a decreasing sequence in Q+ (X) that converges pointwise to the char- acteristic function of A. Then consider the lim sup and lim inf of the sequence (pn(A)).
e. Let (pn) be a sequence in )1t+(X) such that
sup1f dpn < +oo for all f E CC (X).
nEN
(Check that this condition is satisfied if and only if supnEN pn(K) is finite for every compact K of X.)
Prove that the sequence (pn) has a vaguely convergent subsequence.
Hint. Exhaust X by a sequence of compact sets (Kr) and apply Corollary 4.2 on page 19 to each of the separable Banach spaces Cx,,(X).
11. a. Let (fn) be a sequence of increasing functions from R to R such that the series E fn converges pointwise on R to a function f . Prove that the series En o dfn converges vaguely to df (see Exercise 10).
Hint. Consider cp E Cc (R), a compact interval [a, b) in R containing the support of gyp, and a subdivision {xj}o<j<n of [a, b]. Prove that, for every integer I E N,
n-1 +00
w(xj)(f(xj+l) -f(xi)) - II 'II (fk (b) - fk (a))
j=0 k=1+1
I n-i
>2 p(xj)(fk(xj+1) - fk(xj)) k=0 j=0
n-1
>2 (xj)(f(xj+i) - f(xj)).
j=o
b. Example. Let (an) be a sequence in R and (cn) a sequence in R+ such that EnEN en < +oo. Prove that the series of measures &>o cnA , converges vaguely to a positive Radon measure whose distribution function is f = F,n o cnYan, where Ya = l
12. Narrow convergence. We say that a sequence (pn)nEN of positive Radon measures of finite mass on X converges narrowly top E fit j (X) if
An(f) - p(f) for all f E Cb(X).
Every narrowly convergent sequence is vaguely convergent (Exercise 10).
a. A counterexample. Let (an)nEN be a sequence in X with no cluster point. Prove that the sequence (d )nEN does not converge narrowly to 0.
82 2. Locally Compact Spaces and Radon Measures
b. Let p be a positive Radon measure of finite mass and A a Bore]
set whose boundary has p-measure zero. Prove that, if (pn)nEN is a sequence in fit f (X) that converges narrowly to p, then
lim A. (A) = p(A).
n-f+00 Hint. Work as in Exercise 10 above.
c. Let (An) be a sequence in'9Rj (X) and suppose y E9ltl (X). Prove
that the sequence (An) converges narrowly to p if and only if it converges vaguely to it and limn..+0 An (X) = AV) -
Hint. For the "if" part, fix f E Cb (X) and e > 0. Show that there exists a function a E Cc (X) such that a < 1 and f (1 - a) dp < e;
then write
An(f) - p(f) = pn(af) - p(af) + pn((1-a) f) - p((1-a) f).
d. Theorem of P. Levy. If v is a positive Radon measure of finite mass on R, we denote by i the function defined on R by
v(x) _Je%txdv(t).
Let (pn)nEN be a sequence in 9R f (R) and pan element of T? f+ (R).
Prove that (An) converges narrowly to p if and only if the sequence of functions converges pointwise to A.
Hint. Prove that if (àn) converges pointwise to ft, then (f dpn) con- verges to f dp and there exists a dense subspace H in Co (R) such
that
lim f h dpn =
1 fhdp for all h E H n-++00J
(see Exercise 8e on page 42). Conclude with Proposition 4.3 on page 19.
13. a. Let p be a positive Radon measure on X. Suppose the support K of it is compact. Show that there exists a sequence (An) of Radon measures of finite support contained in K that converges narrowly to p (see Exercise 12).
Hint. Take n E N'. Construct a partition of K into finitely many nonempty Borel sets (Kn,P)P<p of diameter at most 1/n. Then, for each p < PN, choose a point xn,p in Kn,p and set
An = E p(Kn,P)bx...y-
P<_Pn
b. Generalize to the case of any positive Radon measure of finite mass.
14. Let g be a Borel function on R taking nonnegative values and locally integrable (see Exercise lb on page 63). Let a be a real number. Consider the function G on R defined by G(x) = fQ g(t) dt.
a. Prove that
ffdG =ff (x)g(x) dx for all f E where dx isLebesgue measure on R.
Hint. If [a, b] is an interval containing the support off and {xj }o<j<n is a subdivision of [a, b], and if we take for each j E {0,... , n - 1} a point tj E [xj, xj+1[, then
n-1 b n-i
f(tj)(G(xj+1)-G(xj))= f f(tj)g(x)dx.
j=0 a j=0
Now use the Dominated Convergence Theorem.
b. Prove that the equality of the preceding question holds when f is any positive Borel function.
15. Recall that fR a-x'dx = f. For all real t > 0, put
r(t) = f+ooxt-to-xdx.
0
Let 8d be the area of the unit sphere in Rd, that is, the mass of the surface measure of the unit sphere in Rd. Prove that 8d = 2ird/2/r(d/2).
Deduce the Lebesgue measure of the unit ball in Rd.
Hint. Compute fkd e-1112dx in two ways.
16. Let a1 be the surface measure of the unit sphere S1 in Rd.
a. Suppose d = 2. Prove that, for any Borel function f from R2 to R+,
2A
if da1 = f f (cos 0, sin 0) d9.
0
Hint. Use polar coordinates.
b. Suppose d = 3. Prove that, for any Borel function f from R3 to R+,
r2A A/2
if dal =
J fA/2f (cos 0 cos W, sin 0 cos gyp, sin W) cos d9 &,p.
o
Hint. Use spherical coordinates.
17. Let a be a positive Radon measure on Rd whose support is contained in the unit sphere Si. Assume a is invariant under orthogonal linear transformations; that is, for any orthogonal endomorphism 0 of Rd and any f E C(Rd),
Jf(Ox) da(x) = Jf(x) da(x).
84 2. Locally Compact Spaces and Radon Measures
a. Show that there exists a function ho from R+ to C such that J e'u-da(y) = h,(IuI) for all u E Rd,
where u y is the scalar product of u and y in Rd. We define h analogously, starting from the surface measure al on S1.
b. Prove that, for all t //E R+,
J ha(tiu[)da,(u) =J rh.,(tiyi)do,(1l),
and so that h,(t) = h (t) (f do,) (f dvl).
c. Deduce that
Hint. Generalize to Rd the result of Exercise id on page 64.
L8. Infinite product of measures, compact case. Consider the space
X=[0,11N={x=(xn)nEN:x,,E[0,1)forallnEN),
and give it the product metric
00
d(x, y) = E 2-nIxn - YnI.
n=0
With this metric, X is compact, by Tychonoff's Theorem. Consider also a sequence (mn)fEN of probabitity measures-that is, Borel measures of mass 1-on [0,1].
a. Show that, for each n E N, the function that maps x E X to xn E [0, 11 is continuous (in fact, Lipschitz).
b. For n E N, denote by Fn the set of functions from X to R of the
form
x H f (x0, ... , xn), with f E CR ([0,1]n+1). Prove the following facts:
i. Fn is a vector subspace of CR(X) for all n E N.
H. Fn C Fn+1 for all n E N.
iii. F = UnEN Fn is a dense vector subspace of CR(X) with the uniform norm [1.1[.
c. For each n, we define a linear form An on Fn by associating to the element
V:xH f(x0,...,xn)
of Fn the real number
An ((P) = 1... rf(x0... n) dmo(xo)...dmn(xn)
Prove that, if V E F, then pp(w) = An(W) for all p > n. Deduce the existence of a linear form p on F such that
p(ip)=pn(S,) for all n E N and W EFn-
Then show that, for <p E F, we have I p(<p) I < II w1I and w > 0 implies p('p) ? 0.
d. Prove that the linear form p extends in a unique way to a positive Radon measure on X.
e. More generally, let (Xn)nEN be a sequence of compact metric spaces and, for each n E N, let mn be a probability measure on X, . Let X = 11nEN Xn be the product space, with the product metric. By working as in the preceding questions, prove that there exists a unique probability measure p on X satisfying)
f(x0i...,xn)dmo(xo)...dmn(xn)=1 f(xo ...,xn)dA(x)
f(^) X
for all n E N and all f E CR(X(n)), where X(n) = [In_o X1. (We thus recover the result of Exercise 5 on page 66 in this particular case.)
19. Haar measure on a compact abelian group. Let X be a compact metric space having an abelian group structure. We assume that addition is continuous as a map from X2 to X.
We denote by B the set of continuous linear forms on CR(X) of norm at most 1. We recall from Exercise 4 on page 20 that B can be given a metric d for which d(pn, p) -+ 0 if and only if
p(f) for all f E C(X),
and that the metric space (B, d) is compact. One can check that the set P of positive Radon measures of mass 1 on X is a nonempty, convex, closed subset of B, and that the topology induced by d on P is that of vague convergence.
a. Markov-Kakutani Theorem. Let K be a nonempty, compact, convex subset of (B, d).
i. Let +p be a continuous affine transformation from K to K (affine means that for any (p, p') E K2 and any a E 10, 11 we have W(ap + (1-a)p') = ap(p) + (1 - a)cp(p')). Prove that cp has at least one fixed point in K - in other words, there is a point A E K such that W(A) = A.
One can work as follows: Let p be any element of K and, for any n EN, set
n
An n+1 Wi(p)
i=O
86 2. Locally Compact Spaces and Radon Measures A. Check that it, E K for each n E N.
B. Let (à,,,,) be a subsequence of the sequence (p,,) that con- verges (with respect to d) to A E K. Prove that, for each integer k, we have (1 + nk) (Apn,,) - E 2B.
C. Deduce that W(A) = A.
ii. Let if be a family of continuous affine transformations of K such that any two elements of if commute. For each cp E if denote by Fp the set of fixed points of V.
A. Prove that all the F, are nonempty, compact, convex subsets of (B, d).
B. Suppose if = (cp,tp'). Prove that cp'(Fw) C F,,. Deduce that ,p and cp' have a common fixed point.
C. Now make no assumption on if. Prove that all the elements of if have at least one common fixed point. (Start with T finite, then use compactness.)
b. For p E 9A+ (X) and x E X we denote by Tx.u the positive Radon measure on X defined by T,,à(f) = f f (x + y) dp(y).
i. Prove that rx(P) C P for all x E X. Deduce that there exists JA EPsuch that rxà=pfor all xEX.
ii. Prove that there exists a Borel measure p on X such that 1A(X)=1
and
f f(t) dp(t) =Jf(x + t) dp(t) for all f E C(X) and X E X.
We call p a Haar measure on X.
c. Uniqueness of Haar measure. Let p and v be Haar measures on X.
Prove that it = v.
Hint. Take f E C(X). Using Fubini's Theorem, compute in two ways the integral
JJf(x+ y) du(x) dv(y).
4 Real and Complex Radon Measures
The framework here is the same as in the previous section. A real Radon measure on X is by definition a linear form p on CR(X) whose restriction to each space CK (X), for K compact in X, is continuous; that is, such that for any compact K of X there exists a real CK > 0 such that
I A(f) I < CK II f II for all f E CK (X ).
We denote by 9RR(X) the set of real Radon measures. We also call the elements of this set linear forms continuous on C R (X ); for an equivalent definition of this notion of continuity, see Exercise 5. By Proposition 3.3, 9)t+(X) C DlR(X). Conversely, every real Radon measure is the difference of two positive Radon measures:
Theorem 4.1 Let it be a real Radon measure on X. For each f E Cc (X), put
à+(f) = sup{à(g) : g E Cc(X) and g < f },
à-(f)=-inf{à(g):gECc (X) and g< f}.
Then à+ and à- can be uniquely extended to positive Radon measures and
à=à+-à .
Proof
1. We first check that the definition of à+(f) given in the statement makes sense. If f E C,,+(X) has support K, then for all g E Cc (X) such that
g<f we havegECK(X),so
11(g):5 11401:5 CKI191I <-CKIIf11.
Thus à+ (f) is well-defined and 0 < à+ (f) < CK 11 f I I. It is also clear that for A real and nonnegative we have à+(Af) = Aà+( f)
2. The essential point is the additivity of à+ on CC (X). Take fl, f2 E C. (X). That à+(f1 + f2) = à+(fl) +à+(f2) will follow from the set equality
{gECC(X):g<-f1+f2}
={gEC,(X):g<f1}+{gECC(X):gSf2}.
One of the inclusions is obvious and the other can be checked quickly:
Suppose g E Cc (X) satisfies g < f, + f2. Put g1 = inf (g, fl) and 92 = 9 - 9i = sup(0, g-f,). We see that 0 < g1 < fl, 0 < g2 S f2, and
9=91+92
3. The same properties hold for à-. On the other hand, if f E CC (X), à+(f) - à(f) = sup{à(9 - f) : g E CC (X) andg < f)
=-inf{à(f-g):gEC.,+(X) and g:5 f}
=-inf{à(h):hECc+ (X) andh< f}=à-(f).
Therefore à(f) = à+(f) - à- (f ).
4. We now extend à+ and à- to CR (X) in the only possible way: Given h E Cc' (X) we take f, g E Cc (X) such that h = f - g (for example, f = h+ and g = h-). Since à+ must be linear on C' (X), we must set
à+(h) = à+(f) - à+(9)
88 2. Locally Compact Spaces and Radon Measures
This definition does not depend on the choice of a decomposition for h.
For if h = f - g' with f', g' > 0, then f + g' = f + g and, by the
additivity of à+ on C,1 (X), we have à+(f) - à+(g) = A+ (f) - p+(g').
One can easily see that the à+ defined in this way is indeed linear and so belongs to fit+(X). We extend à- similarly, and we use item 3 to
show that p = à+ - à-. 0
Remarks
1. The decomposition p = à+ -A- defined in Theorem 4.1 is minimal in the following sense: If p = At - 112 with àl, 112 E fit+(X ), there exists a positive Radon measure v on X such that pi = à++v and p2 = à-+v.
Indeed, it is clear, in view of the definition of p+, that à+(f) < Al(f) for all f E CC (X). One easily deduces from this that the Radon measure on X defined by v = p - à+ is positive. (And of course v = à2 - 14 as well.)
2. Using the same construction, we obtain an analogous decomposition for continuous linear forms on a normed space E that has an order relation
making it into a lattice and satisfying the following conditions, for all f, g E E and all,\ ER+':
- 0< g:5 f implies IIgII 5 11f 11;
- f > 0 implies A f > 0;
- f <gifandonlyifg- f >0.
A bounded real Radon measure on X is by definition a linear form à on CR (X) continuous with respect to the uniform norm on CR (X ); that is, one for which there exists a constant C > 0 such that
IA(f)I <CIIfII for all f ECR(X).
We denote by fit f (X) the set of bounded real Radon measures on X; this is clearly a vector subspace of WIR (X).
Since Ca (X) is dense in the Banach space Co (X) with the uniform norm, every bounded real Radon measure extends uniquely to a continuous linear form on Co (X); this allows us to identify 9)1 f (X) with the topological dual
of COR (X).
Proposition 4.2 Every bounded real Radon measure is the difference of two positive Radon measures of finite mass. More precisely, if p E fit f (X), the Radon measures à+ and p- defined in Theorem 4.1 have finite mass and
IItII=Idà++Idà-,
where IIAII is the norm of p in the dual of Co (X).