ON THE THEORY OF POTENTIALS IN LOCALLY COMPACT SPACES
BY BENT FUGLEDE
Copenhagen
Contents
Introduction ...
Chapter I. Basic concepts of potential theory ...
1. Measures on a locally compact space ...
2. Kernel, potential, energy, capacity ...
Chapter II. The case of a consistent kernel ...
3. The strong topology. ...
4. The interior and exterior capacitary distributions, ...
5. Extensions of the theory ...
Chapter III. Convolution kernels ...
6. Preliminaries concerning locally compact topological groups . .
7. Definite convolution kernels ...
8. Examples. ...
References ...
Page
... 140
... 144
... 144
... 149
... 163
... 164
... 173
... 185
... 188
... 188
... 192
... 204
... 214
Index of terminology and notations Page Page almost everywhere ( = a.e.) . . . . . 146 energy . . . _ . . . 149
balayage . . . 185 energy function . . . . equilibrium distribution capacitable . . . . 157, 163 capacitary distribution . . . . 159, 162 capacity . . . . . 157, 163 concentrated . . . . 146
consistent . . . 167
convolution (SC ) . . . . . . . . 189 ff. convolution kernel . _ . . , _ . . . . 188
definite . . . 151
10 - 603808 Acta mathematics. 103. Imprime le 22 juin 1960 equivalent (measures) ... 166
exterior capacitary distribution .... 183
exterior capacity ... 157, 163 exterior measure ... 146
Haar measure (m) ... 190
interior capaeitary distribution . _ . . . 175
interior capacity . _ . . . . . . . 157, 163 interior measure . . . 146
... 192
... 160
140 B E N T Y U G L E D E Page
K - c o n s i s t e n t . . . 171, 186
K - d e f i n i t e . . . 186
k e r n e l . . . 149
k e r n e l f u n c t i o n . . . 188, i 9 2 K - p e r f e c t . . . 186
m a x i m u m p r i n c i p l e . . . 150
m e a s u r e . . . 144
m u t u a l e n e r g y . . . 149
n e a r l y e v e r y w h e r e ( = n.e.) . . . 15B n e g l i g i b l e . . . 146
p e r f e c t . . . 166
p o s i t i v e . . . 149
9 p o t e n t i a l . . . 149
p r i n c i p l e of c o n t i n u i t y . . . 150
p s e u d o - p o s i t i v e . . . 150
q u a s i - e v e r y w h e r e ( = q.e.) . . . 153
r e f l e x (v) . . . 188
r e g u l a r . . . 150
s t r i c t l y d e f i n i t e . . . 151
s t r i c t l y K - d e f i n i t e . . . 186
s t r i c t l y p o s i t i v e . . . 149
s t r i c t l y p s e u d o - p o s i t i v e . . . 150
s t r o n g ( t o p o l o g y ) . . . 164
P a ~ t r a c e . . . 146
v a g u e ( t o p o l o g y ) . . . 145
w e a k ( t o p o l o g y ) . . . 164
W i e n e r c a p a c i t y . . . 161
a - c o m p a c t , a - f i n i t e . . . 181
91 . . . 154
C, C +, Co, C~- . . . 144
cap, c a p , , c a p * . . . 162
8, 8 + . . . 164
8~, 8}
. . . 187(~-I), (H1), (H2) . . . 180
k (#, ~), /c (x, /~), k (A, /~) . . . 149
k~ . . . 192
K a . . . 144
m . . . 190
' ~ , ' ~ + , '~A, 'm.~ . . . 145 f. u (A), v (A), w (A) . . . 153
u * (A), v* (A), w* (A) . . . 153
U (#), V (~), W (~) . . . 150
/ / ~ , ~ , 1 ~ . . . 155
r A , I ~ . . . 174, 182 e, ez . . . 189 AA, AA, A * , _/k~' . . . . 174 f. 182 f.
/~+, ~ - , ~ S(/~) . . . 146, 144 IIt l . . . 1 8 4
I n t r o d u c t i o n
S i n c e t h e m o d e r n t h e o r y of p o t e n t i a l s w a s i n i t i a t e d b y t h e w o r k s of 0 . F r o s t - m a n [18], M . R i e s z [26], a n d D e l a V a l l S e - P o u s s i n [29], t h e f u r t h e r d e v e l o p m e n t h a s t o a l a r g e e x t e n t c e n t e r e d a r o u n d t h e f o l l o w i n g e s s e n t i a l p o i n t s : (1)
A . T h e d i s c o v e r y b y t I . C a r t a n [10] of t h e f a c t t h a t t h e s p a c e 8 + of a l l p o s i t i v e m e a s u r e s ~ of
finite energy
lI ll ff ix-yF
w i t h r e s p e c t t o t h e N e w t o n i a n k e r n e l i n
R n, n > 2,
is c o m p l e t e i n t h estrong topology,
i.e. t h e t o p o l o g y d e f i n e d b y t h e d i s t a n c e H # - v ] l .
B . T h e s y s t e m a t i c u s e b y J . D e n y [15] of t h e
,Fourier trans/orm
i n t h e s e n s e of L . S c h w a r t z [27] f o r t h e s t u d y of d i s t r i b u t i o n s of f i n i t e e n e r g y w i t h r e s p e c t t o a p o s i t i v e d e f i n i t e d i s t r i b u t i o n k e r n e l i n v a r i a n t u n d e r t r a n s l a t i o n s i n R n.(1) As to these and f u r t h e r lines of research, see the expository article on m o d e r n p o t e n t i a l t h e o r y b y M. Brelot [7].
O N T H E T H E O R Y O F P O T E N T I A L S I N L O C A L L Y C O M P A C T S P A C E S 141 C. The study of potentials of measures on a locally compact space with respect to a
regular
kernel (i.e. a kernel satisfying the principle of continuity, cf. w 2.1). The main tool is here thevague topology,
which is related to the classical notion of con- vergence for positive measures. See G. Choquet [12, 13], M. Kishi [19], N. Ninomiya [23], M. Ohtsuka [24, 25].D. The theory of abstract
capacities
developed b y G. Choquet [14].The present exposition is devoted to the theory of potentials of measures on a locally compact space, in particular on a group, the main emphasis being placed upon the study of
capacity
andcapacitary distributions.
I n the existing literature concerning this subject the more advanced p a r t of the theory is based on the assumption t h a t the kernel be regular (cf. the references under C above), and it is usually required t h a t the kernel fulfil F r o s t m a n ' smaximum principle.
I n s t e a d of making assumptions of this nature we have chosen to base our s t u d y on the strong topology as well as the vague topology. Two papers b y t I . Cartan [9], [10] have served as a guide. We shall investigate the case of a positivede/iuite
kernel possessing the following two prop- erties: (i) the space E + is strongly complete (cf. above under A), and (ii) the strong topology on ~+ ist stronger ( = f i n e r ) t h a n the vague topology on ~+. A positive de- finite kernel possessing these two properties will be calledper/ect.
(1) I t turns out t h a t the desired t y p e of results concerning eapacitary distributions (associated with a r b i t r a r y sets) and capacitabflity of analytic sets can be obtained in a simple a n d natural w a y in case of a perfect kernel (or just a consistent kernel). (1) These results are of a global nature, unlike the corresponding results based on the m a x i m u m principle a n d the vague topology. This fact reflects theglobal character
of the concept of a perfect or a consistent kernel (cf. w 7.3). A p a r t from its global character, the concept of a con- sistent kernel is, however, more general t h a n t h a t of a kernel fulfilling F r o s t m a n ' s maxi- m u m principle. This becomes clear if we consider the case of acompact
space (whereby the global aspects disappear). Then a n y positive kernel/c satisfying the m a x i m u m prin- ciple is positive definite (Ninomiya [23]) and regular (Choquet [12]), and hence it follows from the proof of a theorem due to M. Ohtsuka [24] t h a t /c is consistent (cf.w 3.4 of the present paper).
The fact t h a t the Newtonian kernel (and, more generally, the classical Green's function) is perfect, was proved b y H. Cartan [10]. I t is also known t h a t the kernels (1) A perfect kernel is, in particular,
strictly
definite, that is, the energy of a measure/t :~ 0 is> 0 if at all defined. This strict definiteness (the so-called principle of energy) is, however, of minor importance for the development of the theory, and we have therefore introduced the weaker concept of a
consistent
kernel (w 3.3). Such a kernel is definite, but not necessarily strictly definite.1 4 2 BENT FUGLEDE
I x - y ] ~ ~ of order a in R n are perfect ( 0 < g < n ) , whereas the m a x i m u m principle is fulfilled for ~ < 2 only. This perfect character of the kernels of order ~ (among other kernels) was established first b y J. Deny [15, 16]. (1) We present an alternative, more elementary and more direct proof based exclusively on M. Riesz' composition formula (cf. Theorem 7.4 and w 8.1). The first result in this direction is due to H. Cartan [9], who proved t h a t the kernels of order a are K-perfect, t h a t is, perfect when considered only on compact subsets of the space. Actually, a n y strictly positive definite con- volution kernel has this latter property, as we shall s h o w in w 7. Altogether it t m ~ s out t h a t practically all the definite kernels usually m e t with in analysis are consistent (and hence perfect if t h e y are strictly positive definite).
The contents of the present p a p e r m a y be summarized as follows.
Chapter I is of a p r e p a r a t o r y character, and the methods and m o s t of the results are well known. After a brief survey over the relevant parts of the theory of meas- ures and integration on a locally compact space X (w 1) follows an exposition of the theory of capacity and of the capacitary distributions(2) on compact sets (w 2). I n this section the potential and energy of measures are formed with respect to an arbitrary kernel on X. (A kernel on X is defined as a lower semi-continuous function k = k (x, y) on X • The proofs are based on the facts t h a t potential a n d energy are lower semi-continuous functions on the space of all positive measures on X with the vague topology, and t h a t the class of all positive measures of total mass 1 supported by" a compact set is c o m p a c t in the vague topology.
I n Chapter I I the kernel k is supposed to be positive de/inite. (We usually omit the qualification "positive".) The class E of all measures # (of variable s i g n ) o f finite energy
(1) I n D e n y ' s t h e o r y , r e f e r r e d to a b o v e u n d e r B, is c o n t a i n e d t h a t a wide class of p o s i t i v e de- f i n i t e c o n v o l u t i o n k e r n e l s o n R n h a v e p r o p e r t i e s v e r y s i m i l a r to t h o s e of a p e r f e c t k e r n e l in o u r s e n s e (cf. D e n y [15], C h a p . I, 3), t h e sole m o d i f i c a t i o n b e i n g t h a t t h e c o n c e p t of e n e r g y h a d to b e d e f i n e d in a m a n n e r q u i t e d i f f e r e n t f r o m t h e classical d e f i n i t i o n as a L e b e s g u e i n t e g r a l a d o p t e d in t h e p r e s e n t p a p e r . I n a s u p p l e m e n t a r y p a p e r D e n y [16] p r o v e d t h a t t h i s d i f f i c u l t y c a n b e o v e r c o m e u n d e r t h e a d d i t i o n a l a s s u m p t i o n t h a t t h e k e r n e l be r e g u l a r . N e v e r t h e l e s s , it s e e m s fair to s a y t h a t t h e m e t h o d s o f F o u r i e r a n a l y s i s a r e n o t r e a l l y a d e q u a t e i n t h e finer s t u d y of c a p a c i t y a n d c a p a c i t a r y d i s t r i b u t i o n s . (2) I f t h e k e r n e l fulfills F r o s t m a n ' s m a x i m u m p r i n c i p l e , t h e c a p a c i t a r y d i s t r i b u t i o n s a r e also called e q u i l i b r i u m distributions b e c a u s e t h e i r p o t e n t i a l is c o n s t a n t in t h e s e t in q u e s t i o n ( e x c e p t in s o m e s u b s e t of zero c a p a c i t y ) . I f t h e k e r n e l is s t r i c t l y d e f i n i t e , t h e r e is j u s t o n e c a p a c i t a r y distri- b u t i o n o n a g i v e n set.
(3) T h e m o s t i m p o r t a n t c a s e is t h a t of a p o s i t i v e k e r n e l . O<~k(x, y)<~ + c~. W e also a d m i t k e r n e l s of v a r i a b l e s i g n , b u t in t h a t case we r e s t r i c t t h e a t t e n t i o n to p o t e n t i a l s a n d e n e r g y of m e a s - u r e s of ( u n i f o r m l y ) c o m p a c t s u p p o r t s .
O ~ T H E T H E O R Y O F P O T E N T I A L S I N L O C A L L Y C O M P A C T S P A C E S 143
is a p r e - H i l b e r t s p a c e w i t h t h e e n e r g y n o r m [[ # ]] = (k (/~, #)) }. T h e s u b s e t E + f o r m e d b y all
positive
m e a s u r e s in E is a c o n v e x cone. A m o n g t h e definite k e r n e l s we single o u t t h o s e for which t h e t w o t o p o l o g i e s ( s t r o n g a n d vague) on E + h a v e t h e following p r o p e r t y of consistency: I f a s t r o n g C a u c h y f i l t e r 9 c o n v e r g e s v a g u e l y t o some m e a s - ure /~0, t h e n 9 s t r o n g l y . A definite k e r n e l w i t h t h i s p r o p e r t y will be calledconsistent.
I t is e a s i l y shown t h a t a k e r n e l is p e r f e c t if a n d o n l y if i t is c o n s i s t e n t a n d s t r i c t l y definite. C e r t a i n sufficient c o n d i t i o n s of a general n a t u r e a r e o b t a i n e d , a n d i t is shown t h a t a k e r n e l o b t a i n e d b y s u p e r p o s i t i o n of c o n s i s t e n t k e r n e l s is con- sistent. U n d e r t h e h y p o t h e s i s t h a t t h e k e r n e l b e c o n s i s t e n t we p r o c e e d t o i n t r o d u c e t h einterior and exterior capacitary distributions
a s s o c i a t e d w i t h a n a r b i t r a r y set of finite i n t e r i o r , resp. e x t e r i o r , c a p a c i t y . (1) W e follow t h e m e t h o d i n d i c a t e d b y H . Car- t a n [10], w 6, for t h e N e w t o n i a n k e r n e l (cf. also A r o n s z a j n & S m i t h [1] for t h e k e r n e l s of o r d e r ~.), b u t c e r t a i n m o d i f i c a t i o n s arc r e q u i r e d u n d e r t h e p r e s e n t g e n e r a l circum- stances. A s a b y - p r o d u c t we o b t a i n t h e following p r o p e r t y of t h e e x t e r i o r c a p a c i t y of a r b i t r a r y s e t s :cap* A = l i m cap* A~
n
for a n y i n c r e a s i n g sequence of s u b s e t s
A ~ X .
This r e s u l t is t h e k e y t o a n a p p l i c a - t i o n of C h o q u e t ' s t h e o r y r e f e r r e d t o a b o v e u n d e r D, a n d we c o n c l u d e t h a t e v e r y K - a n a l y t i c s u b s e t of X is c a p a c i t a b l e , i.e. of e q u a l i n t e r i o r a n d e x t e r i o r c a p a c i t y . (2) T h e c h a p t e r ends w i t h a brief discussion of t h e t h e o r y of " b a l a y a g e " .C h a p t e r I I I is d e v o t e d t o t h e p a r t i c u l a r l y i n t e r e s t i n g case in which t h e space X is a l o c a l l y c o m p a c t t o p o l o g i c a l
group
a n d t h e k e r n e l a c o n v o l u t i o n kernel, i.e.k(x, y)=k(xy-1),
(1) In the study of exterior capacitary distributions we must impose upon the locally compact space X a certain restriction, e.g. that X be metrizable.
(-") For the Newtonian kernel, as well as for Green's function, this result was obtained by G. Chequer [14], Chap. II, by application particularly of Cartan's maximum principle (tt. Cartan [10], w VI), which leads to the fundamental inequality
cap (A [ J B ) + c a p (A N B ) < c a p A + c a p B
for arbitrary compact sets A and B. The method described above in the text was used first by Aronszajn and Smith [1] in ease of the kernels of order a. Recently, 1~[. Kishi [19] has established the capaeitability of all relatively compact Borelian or K-analytic subsets of a locally compact space of which every compact subset is metrizable, the assumption on the kernel being closely related (in fact equivalent) to Frostman's maximum principle.
144 BENT FUGLEDE
t h e " k e r n e l f u n c t i o n " k = k ( x ) being a g i v e n lower s e m i - c o n t i n u o u s f u n c t i o n on t h e g r o u p X . I t is shown t h a t a n y d e f i n i t e c o n v o l u t i o n k e r n e l is K - c o n s i s t e n t ( T h e o r e m 7.2). T h e p r o o f is b a s e d on t h e f a c t t h a t t h e energy /unction
is b o u n d e d a n d u n i f o r m l y c o n t i n u o u s p r o v i d e d /~ E ~+ ( T h e o r e m 7.1). This r e s u l t fol- lows, in t u r n , f r o m a l e m m a a s s e r t i n g t h a t a p o s i t i v e definite, lower s e m i - c o n t i n u o u s f u n c t i o n / o n a t o p o l o g i c a l g r o u p is b o u n d e d a n d u n i f o r m l y c o n t i n u o u s p r o v i d e d / i s f i n i t e a t t h e i d e n t i t y . T h e p r o p e r t y of a definite c o n v o l u t i o n k e r n e l t o be a c t u a l l y c o n s i s t e n t (not o n l y K - c o n s i s t e n t ) d e p e n d s , therefore, on t h e b e h a v i o u r of t h e k e r n e l f u n c t i o n k (x) as x t e n d s t o i n f i n i t y in X , cf. w 7.3. I n case of a n A b e l i a n g r o u p we p r o v e , in p a r t i c u l a r , t h a t if k = k (x) h a s t h e f o r m
w h e r e h~>0 is lower s e m i - c o n t i n u o u s on X , t h e n t h e c o n v o l u t i o n k e r n e l Ic(xy -1) is consistent, a n d ~+ is c o m p l e t e ( T h e o r e m 7.4). S e v e r a l i m p o r t a n t kernels, i n c l u d i n g t h o s e of o r d e r a, a r e of t h i s form. S o m e t y p e s of k e r n e l s s t u d i e d b y K . K u n u g u i [20], N. N i n o m i y a [22], a n d T. U g a h e r i [28] are i n v e s t i g a t e d f u r t h e r (w 8.2), a n d t h e p a p e r closes w i t h a n u m b e r of e x a m p l e s s e r v i n g to i l l u s t r a t e v a r i o u s p o i n t s in t h e t h e o r y .
I . B A S I C CONCEPTS OF P O T E N T I A L T H E O R Y 1. Measures on locally compact spaces
T h e d i s t r i b u t i o n s of m a s s (or charge) to be considered in t h e p r e s e n t s t u d y a r e t h o s e w h i c h can be i n t e r p r e t e d m a t h e m a t i c a l l y as r e a l - v a l u e d measures, in p a r t i c u l a r p o s i t i v e m e a s u r e s . R e f e r r i n g t o N . B o u r b a k i [4], [5] for a n e x p o s i t i o n of t h e t h e o r y of m e a s u r e s a n d i n t e g r a t i o n on a l o c a l l y c o m p a c t H a u s d o r f f space, we l i m i t ourselves t o l i s t i n g (in w 1.1) t h o s e c o n c e p t s w h i c h a r e e s p e c i a l l y r e l e v a n t in p o t e n t i a l t h e o r y , a n d t o s t a t i n g (in w 1.2) s o m e f u r t h e r n e c e s s a r y results. A s t o t h e t e r m i n o l o g y a n d n o t a t i o n s we g e n e r a l l y follow B o u r b a k i . (1)
(1) Certain exceptions from this convention will be listed here. Our locally compact (I-Iausdorff) space is denoted by X, and the class of all continuous functions on X by C = C (x). (When speaking of a continuous function, we generally understand that the values are ]inite real numbers. A lower semi-continuous function is allowed to take the value + c~, but not - c~.) The class of all continuouS functions of compact support is denoted by Co = Co (x). :For any class :~ of functions, 5 + denotes the class of all positive functions (/>~ 0) from 5. The support of a function, or a measure, is denote d by S (]), resp. S (/~). A set A c X is said to be of class K a if A may be represented as the union o f some sequence of compact subsets of X.
ON T H E T t t E O I ~ Y OF P O T E N T I A L S I N L O C A L L Y COMPACT S P A C E S 145 1.1. Principal notions. The class of all (Radon) measures on a locally c o m p a c t (Hausdorff) space X is denoted b y ~ = ~ (X). On this linear space, the semi-norms
~ - - ,
f/d~, /eCo,
define t h e so-called vague topology. A filter q5 on ~ (cf. Bour- baki [2], Chap. I, w 5) converges v a g u e l y to ~0 E ~ iflira f ] d/~ = f / d it o along (I) for e v e r y / E Co.
A subclass B of 7tl is called v a g u e l y b o u n d e d if each of the a b o v e semi-norms re- mains b o u n d e d on B. A n y v a g u e l y b o u n d e d subclass of ~ is relatively compact in t h e v a g u e t o p o l o g y (Bourbaki [4], Chap. I I I , w 2, N ~ 7). The converse is obvious since the semi-norms are continuous. P a r t i c u l a r l y useful is the induced v a g u e t o p o l o g y on
~ + = ~ + (X), t h e class of all positive measures. The space ~ + is v a g u e l y complete, a n d hence closed in ~ . YIore generally, t h e class ~ of all positive measures sup- p o r t e d b y a given closed set F is a closed convex cone in ~ .
T h e integral (strictly speaking: u p p e r integral) of a lower semi-continuous function g>~ 0 with respect to a measure # >~ 0 is defined b y
f gd~=
supfide.
f 9 c +,f<
This integral is additive a n d positive homogeneous in g a n d /z. Clearly, the m a p p i n g -->fgd/~, # E ~ +, is lower semi-continuous in the v a g u e t o p o l o g y on ~ + for fixed lower semi-continuous g~>0. (1) Since the characteristic function ~ associated with an open set G c X is lower semi-continuous, we m a y define t h e measure of G b y
~(G)= f ~od~.
S u b s e q u e n t l y one introduces the class s (/z) of ~-integrable functions / with val- ues - ~ < / ( x ) ~ < + ~ (Bourbaki [4], Chap. IV, w 1 6 7 The integral of / with re- (1) Under the additional assumption that /~ E ~ + be of compact support, one obtains a de- finition of f gdl~ for any lower semi-continuous g (whether positive or not) by replacing the class C~
by the class Co in the above definition. It is easily verified that this integral is additive und positive homogeneous with respect to the two variables g and r Denoting by c~> 0 a constant such that g (x)/> - c everywhere in S (/z), we get
f g d ~ f (g§ dt,.
compact, the mapping # -* r J gd/z is lower semi-continuous on ~ + for any given lower If X is
semi-continuous function g on X.
1 4 6 BENT FUGLEDE
spect to # is d e n o t e d b y
f/d/u.(1)
Taking [ = ~ v A = t h e characteristic function asso- ciated with a set A c X , we o b t a i n the class of #-integrable sets. T h e measure of such a set is defined b y /U ( A ) = f ~oAd/U. An3; c o m p a c t set is integrable. Next, one defines t h e local concepts of a~a-measurable
function a n d a /a-measurable set (Bourbaki [4], Chap. IV, w 5).F o r an a r b i t r a r y set A c X the
exterior
a n d theinterior measure
of A are deter- mined b y/U (A) = inf/U (G) a n d /U. (A) = sup/U (K), (1)
G D A K ~ A
respectively. (The letter G refers to open sets a n d K to c o m p a c t sets.) The e q u a t i o n
/U*(A)=/z,(A)
subsists n o t a b l y if A is open, or if A is /u-measurable a n d c o n t a i n e d in the union of some sequence of /u-integrable sets. A set N c X is called~u-negligible
if /U* (N) = 0, a n d locally /u-negligible if N f3 K is /u-negligible for every c o m p a c t set K.
These concepts lead to the notions
~u-almost everywhere
(/u-a.e.) a n dlocally ~u-almost everywhere,
respectively.The
trace /UA
of a measure /U>~0 on a /u-measurable setA c X
is defined b y /aA = ~0a 9 i.e.,f /d/uA= f /.wd/u, /eC~. (2)
(Observe t h a t /~vA is #-integrable.) T h e total mass of/UA i s / U A ( X ) = / U . ( A ) (cf. L e m m a 1.2.2 below). A measure /U ~> 0 is said to be
concentrated
on a set A if t h e comple- m e n t C A is locally /u-negligible; or, equivalently, if A is /a-measurable and/U =/UA. I t follows t h a t /U(X)=/U.(A). (If A is closed, or if for instance # * ( X ) < + c ~ , t h e n/U*(CA)=/a.(CA)=O,
t h a t is, C A is /u-negligible. A measure # is, therefore, concen- t r a t e d on aclosed
set A if a n d only] if /U issupported
b y A, which m e a n s t h a tS(/U)cA.)
F o r a n y measure /U>~0 a n d a n y /u-measurable set A, /UA is c o n c e n t r a t e d on A. W e denote b y ~ t h e class of all positive measures c o n c e n t r a t e d on a given setA c X .
Using t h e canonical decomposition / u = # + - / a - , one defines m e a s u r a b i l i t y a n d integrability with respect t o a measure /a of
variable sign
b y requiring m e a s u r a b i l i t y a n d integrability with respect to /~+ a n d /U-, or, equivalently, with respect to I/U]=/U + +/U-. F o r a n y /u-integrable function [ one defines(1) In particular, a lower semi-continuous function g ( ~> 0 unless S (/x) is compact) is integrable respect to a measure /x~> 0 if and only if J
gd/~,
as defined above, is finite. :In the affirmativew i t h
case the two notions of integral coincide.
O N T H E T H E O R Y O F P O T E ~ q T I A L S I ~ L O C A L L Y C O M P A C T S P A C E S 147
The same formula serves to define the integral of a lower semi-continuous function / provided
f / d # +
andf / d # -
are not both infinite; moreover, it is assumed either t h a t f~>0 or t h a t S(/~) is compact (el. note 0), P. 145).The
product ,a | v
of two measures # and v on the locally compact spaces X and Y, respectively, is defined as the unique measure on the product space X • Y such t h a tfor every
qECo(X), ~ECo(Y).
Here ~0| denotes the functionep(x).~(y)
of class Co (X • Y). When integrating with respect to # | v one m a y writef f r
(x, y) d/~ (x)dv (y)
instead off ] d ( ~ Q v ) .
The following two instances of Fubini's theoremf f / ( x , y ) d t ~ ( x ) d v ( y ) = f d t ~ ( x ) f / ( x , y ) d v ( y ) = f d v ( y ) f / ( x , y ) d t t ( x )
(3) will be used repeatedly in the sequel:(i) I f / is
integrable
with respect to # | i.e. / E s | then the interior integrals on the right represent functions Of class El(#) and s (v), respectively, de- fined and finite almost everywhere, and (3) subsists.(ii) If / is
lower semi-continuous
(and ~>0 unless tt and v have compact sup- ports), then (3) holds provided the integral with respect to # | v is defined. (Cf. above.See also L e m m a 1.2.6 below.) This is always the case if /~>~ 0 and v~> 0, in which case the interior integrals on the right represent lower semi-continuous functions of x and y, respectively.
As to the proofs, see Bourbaki [5], w 8, N ~ 1, for the case of positive measures;
and a p p l y L e m m a s 1.2.3 and 1.2.6 below in the general case.
1.2.
Supplementary results.
I n order to sav e space we omit the proofs of the following lemmas.LE~MA 1.2.1.
I/ a locally compact space X is metrizable and o/ class Ko, then
~ (X) satis/ies the /irst axiom o/ countability.
I n view of this l e m m a the use of filters m a y often be avoided in the sequel if one assumes t h a t the locally compact space X is metrizable and of class K~ (or, equivalently, t h a t X satisfies the 'second axiom of countability, cf. Bourbaki [3], w 2,
148 B E N T F U G L E D E
N ~ 9). I n case of such a space X, a positive measure if adheres to a subset S ~ ~ + (X) if a n d only if S contains a sequence converging t o ft. I n particular, g is closed if a n d only if $ is sequentially closed. Similarly, a n y b o u n d e d sequence in ~ + contains a convergent subsequence.
W e r e t u r n to the s t u d y of measures on an a r b i t r a r y locally c o m p a c t space X.
LEMMA 1.2.2. For any lower semi-continuous function g>~O, any measure f f ~ O , and any if-measurable set A ~ X ,
fgdif = f gdif (K compact).
This is a generalization of t h e second p a r t of (1), w 1.1, which corresponds to t h e case g = 1. T h e following l e m m a deals with measures of a r b i t r a r y sign.
L E M M A 1.2.3. Suppose g is lower semi-continuous (and />0 unless # a n d v have c o m p a c t supports). The identity
f gd (aif + b )=a f g @ + b f gdv
subsists in the sense that the integral on the left is defined whenever the two integrals on the right are both defined and the linear combination is meaningful.
The remaining~ three lemmas are concerned with the p r o d u c t of two measures on t w o locally c o m p a c t spaces X a n d Y, respectively.
LEMMA 1.2.4. The mapping (if, v ) - - > f | o/ ~ + ( X ) x ~ + ( Y ) into ~ + ( X x Y ) is continuous.
(Cf. B o u r b a k i [4], exerc. 5, p. 100). T h e corresponding assertion concerning meas- ures of a r b i t r a r y sign would be false.
L E M M A 1.2.5. I f A c X and t B ~ Y are measurable with respect to I f E T R + ( X ) and v E ~ + (Y), respectively, then the trace o/ if | v upon A x B equals ifA @ VB.
I n particular, f f | is c o n c e n t r a t e d on A x B if ff is c o n c e n t r a t e d on A a n d o n B .
L E M M A 1.2.6. The following identities hold /or any two measures ff E ~ (X) and
S (if | ~) = S (if) x S (~),
lif | |
(,a | v) + = (,a + | v +) + (if- | v-),
( f f | | - |
O N T H E T H E O R Y O F P O T E N T I A L S I N L O C A L L Y C O M P A C T S P A C E S 149
2. K e r n e l , potential, e n e r g y , c a p a c i t y
2.1.
De/initions.
B y akernel
on a locally c o m p a c t (Hausdorff) space X is m e a n t a lower semi-continuous functionk = k ( x , y ) ,
defined everywhere inX •
with values- oo < k ( x , y ) < + ~ .
A kernel is called
positive
ifk (x, y)>~ 0
for e v e r y pair(x, y),
a n dstrictly positive
if,in addition, k ( x , x ) > 0 for every x E X . A kernel k is called
symmetric
if k ( x , y ) == k(y, x).
F o r a n y kernel 2, t h e functions k(x, 9 ) a n d k ( . , y) are lower semi-contin- uous in X for fixed x a n d y, respectively.W e proceed to define p o t e n t i a l a n d e n e r g y of measures with respect to a given kernel k. I n order to uvoid certain difficulties
we shall always assume that the meas- ures in question have compact supports, except in the case o/ a positive kernel,
where a r b i t r a r y measures are a d m i t t e d . Thepotential
of a measure # on X at a point x E X is t h e n defined b yk(x,
Et)=S k(x, y)d/~(y)=k(x, # + ) - k ( x , #-)
provided
k(x,
#§ a n dk(x, #-)
are n o t b o t h infinite. I n particular, the potential of a positive measure is defined everywhere a n d represents a lower semi-continuous func- tion on X. W e shall sometimes use the n o t a t i o nk (A, # ) = sup k (x,//,)
x e A
( A c X ; #>~0).
The
mutual energy of
two measures ~u a n d v (of c o m p a c t supports unless k~>0) is defined b yk ( # , v ) =
f f k ( x , y)d/~(x)dv(y)=k(#+,v+)+k(# -,
v - ) - k ( / ~ +, r - ) - k ( # - , v +) (1) p r o v i d e d k ( / ~ + , v + ) + k ( # - , vTM)
ork ( # + , v - ) + k ( # - , v +)
is finite (ef. L e m m a 1.2.6);thus in p a r t i c u l a r if /~ >~ 0 a n d v>~ 0. F r o m F u b i n i ' s t h e o r e m follows t h a t k ( # , v ) =
f k ( x ,
v ) d # ( x ) =fk(/~, y)d~(y)
whenever k (~u, v) is defined. F o r v = # we o b t a i n t h e
energy
of /~:k ( # , # ) =
f f k(x, y)d/~(x)d/~(y)= S k(x, #)d/~(x).
If the kernel k is
symmetric,
we have the law of reciprocity150 B E N T F U G L E D E
/ k (x,/t) du (x) = f k (x, v) d / t (x), valid a t least w h e n k (it, v) is defined.
The following three functions defined for positive measures tt (of c o m p a c t sup- p o r t unless /c~> 0) present a certain similarity, a n d each of t h e m gives rise to a con- cept of c a p a c i t y (cf. w 2.3):
U (~) = k ( x , ~) = sup ~ (x, ~)
x e X
v (~) = k ( s (~), ~) = sup k (x, ~)
x e s ( t o
Clearly,
W ( ~ ) = k ( ~ , ~) = . I k ( x , ~ ) d ~ .
- ~ < w ( ~ ) < v ( ~ ) . t , ( x ) ; v ( ~ ) < u ( ~ ) .
A kernel is said to satisfy F r o s t m a n ' s
maximum principle
if U ( / ~ ) = V(/~) for every E ~ + of c o m p a c t support. (1) (If a positive kernel satisfies this m a x i m u m principle, t h e n U (/~) = V (~u) forevery /~ E ~+.)
A kernel k will be called
pseudo-positive
if W (ju)/> 0 for e v e r y /~ E ~ + of com- p a c t support; a n dstrictly pseudo.positive
if W(/~)> 0 for e v e r y ju E ~ + , # ~ : 0 , of com- p a c t support. A n y positive kernel is pseudo-positive, a n d it is strictly pseudo-positive if a n d only if it is strictly positive. (In fact, if k(/~,/~) = 0 , t h e open set of pairs (x, y) such t h a t k (x, y) > 0 does n o t m e e t t h e s u p p o r t S (/~) • S (/~) of # | H e n c e k (x, x) = 0 for every x E S (/~). T h e converse s t a t e m e n t is verified b y t a k i n g /~ = ex ( = t h e mass § placed a t the point x.)A kernel /c is called
regular
if it satisfies theprinciple o] continuity,
i.e. if one can conclude t h a t t h e potential ]c (x,/~) of a measure /~>~ 0 of c o m p a c t s u p p o r t is continuous t h r o u g h o u t X when it is k n o w n t h a t the restriction ofk(x, [~)
to S ( # ) i s continuous. (2) As to t h e s t u d y of potentials with respect t o a regular kernel, or a kernel satisfying F r o s t m a n ' s m a x i m u m principle, see for instance the literature referred t o in the i n t r o d u c t i o n (under C). I n t h e present s t u d y we shall generally n o t m a k e assumptions of this n a t u r e (cf., however, Theorems 3.4.1, 3.4.2, a n d 7.3).(1) The fact that the :Newtonian kernel satisfies this maximum principle was proved by M. A. J.
Maria [21]. The kernels of orders ~< 2 have the same property, as shown by Frostman [18], p. 68.
(2) The regularity of the Newtonian kernel was proved by G. C. Evans [17], p. 238, and by F.
Vasilesco [31]. For the kernels of order a, 0 < ~ < n , see Frostman [18], p. 26. More generally-, S.
Kametani has established the regularity of any kernel on
R n
which is a continuous, decreasing, and positive function of ] x - y l {cf. K. Kunugui [20], p. 78). A further sufficient condition for regularity is found in H. Cartan & g. Deny [lI], w167 6, 7.ON T H E T H E O R Y OF P O T E N T I A L S I N L O C A L L Y COMPACT S F A C E S 151 I n the sequel we shall concentrate on either of the following two cases:
I: The kernel k is positive: k(x, y)~>0.
I I : The space X is compact.
The remaining case Of a kernel of variable sign on a locally compact, non compact space presents certain difficulties unless the a t t e n t i o n is limited (as described above) to meas- ures supported by some (fixed) compact subset K c X (cf. w 5.3). This limitation actually amounts to replacing X b y the compact space K , and k b y its restriction to
K •
so t h a t one is back in Case I I . Throughout the rest of Chapter I and the whole of Chapter I I (except for w 5.3) we shall therefore always assume t h a t one (or b o t h ) o f the above cases I or I I occurs. This general hypothesis will usually not be repeated.Case I I can mostly be reduced to Case I simply b y replacing the kernel k b y the positive kernel k' obtained b y adding to k a suitable constant c>~0:
k' (x, y)=k(x, y)+c>~O.
This is always possible since a lower semi-continuous function is bounded from below on a compact space.
A kernel ]c is called
de/inite
( = positive definite) if it is symmetric and if the energy k (#, #) is ~> 0 whenever defined; andstrictly de]inite
if, in addition, k (/~, #) = 0 implies # = 0. Thus a symmetric kernel is definite if a n d only ifk ( ~ +, ~ + ) + k (~-, ~-)~> 2 k (~ +, ~ - )
for every measure /~. A n y definite kernel is pseudo-positive, and a n y strictly definite kernel is strictly pseudo-positive. Chapters I I and I I I are devoted to the s t u d y of potentials with respect to a definite kernel.
I n the remaining p a r t of Chapter I, only
positive measures
will be considered, and the space 71~ + of all such measures will be t h o u g h t of as a Hausdorff space with the vague topology (w 1.1).(a)
(b) (c) (d) (e)
2.2.
Potential and energy o/ positive measures.
LEMMA 2.2.1.
The /ollowing live /unctions are lower semi-continuous:
k(#, v ) =
f f k ( x , y)d/~(x)gv(y) on
~ § 2 1 5 +.k ( x , # ) = f k(x, y) d#(y)
on X x ~ +.v ( ~ ) = ~ ( x ,
~) on ?~+.
V (~) = k ( s (~), ~) on ~ + .
W(/~) =k(~, ~)
on ~ + .152 ~E~T FUGLEDE
Proo/. A d (a): consequence of L e m m a 1.2.4 since t h e function f k d 2 of ~ E
~ + ( X • is lower semi-continuous (w 1.1). A d (b): consequence of (a) because t h e map- ping x--> ex of X into ~ + is continuous a n d k (x, # ) = k (ex,/t) when ex denotes t h e mass + 1 placed at t h e p o i n t x E X. A d (c) a n d (d): consequences of (b). W e show this in t h e case of the function V(/t). L e t /t0 E ~ + , t < V(/t0 ). T h e n k(x0,/t0) > t for some x 0 ES(~o). I n view of (b) there are n e i g h b o u r h o o d s A of x o in X a n d B of/~0 in ~ + such t h a t
k ( x , / t ) > t for x E A , / ~ E B . (I) Since x o E S (/to), there is a function / E C~ with S (]) c A such t h a t f / d/~0 4: 0, a n d hence f / d / t : 4 : 0 for every /t in some n e i g h b o u r h o o d B' of /t 0. This implies t h a t S ( # ) has some p o i n t x in c o m m o n with A when # E B'. Using this p o i n t x in (1), we con- clude t h a t V (/t) >~ k (x, #) > t for every I ~ E B N B'. A d (e) : consequence of (a).
L E M ~ A 2.2.2. I / a positive measure ~u is concentrated on some set A ~ X , then U (/~) = lim U (/tK); V (/t) = lim V (/tK); W (/t) = lim W (/tK),
K t a K t A ~ t A
where #K denotes the trace o/ # upon K, and K ranges over the increasing /iltering /amily o/ all compact subsets o/ A . (1)
Proo/. According to L e m m a 1.2.2. with g E C~, #K ->/tA = # v a g u e l y as K t A.
H e n c e it follows f r o m t h e preceding l e m m a t h a t U (/~) ~< lim U (/~K) ~< U (#)
K~A
(in Case I), a n d similarly w i t h V or W in place of U. Case I I (X c o m p a c t ) f o l l o w s f r o m Case I in t h e usual way.
Remark. If # >~ 0 a n d v/> 0 are c o n c e n t r a t e d on A c X a n d B c X, respectively, it follows in t h e same w a y t h a t
k (tt, v ) = lim k (/tH, V)= lim k (/~, vK)= lim k (/~H, V~),
z l t A K~B ~ t ~ , K t B
where H a n d K d e n o t e a r b i t r a r y c o m p a c t subsets of A a n d B, respectively.
(1) As to the concept of a filtering family (generalizing that of a monotone sequence), we refer to Bourbaki [2], Chap. I, w 5, N ~ 4. In case I (k>~0) the sign "lira" may be replaced by supremum over all compact subsets K c A.
O N T H E T H E O R Y O F P O T E N T I A L S I N L O C A L L Y COMPACT S P A C E S 153 2.3. Capacities associated with a kernel. Given a kernel k = k ( x , y ) o n X , we derive from each of t h e functions U(/~), V(~u), W ( # ) a set function b y defining, for a n y set A c X ,
u ( A ) = i n f U(#); v ( A ) = i n f V(#); w ( A ) = i n f W(#), (1) where F, ranges over t h e class of all positive measures c o n c e n t r a t e d on A a n d of t o t a l mass # ( X ) = / ~ ( A ) = 1. (We i n t e r p r e t these infima as + ~ if A is void.) The result would be t h e same if S(/~) were required to be c o m p a c t a n d contained in A; this follows easily f r o m L e m m a 2.2.2 a n d t h e second relation (1), w 1.1. H e n c e
u ( A ) = i n f u ( K ) ; v ( A ) = i n f v(K); w ( A ) = i n f w ( K ) ,
(2)
where K ranges over the class of all c o m p a c t subsets of A. Clearly, each of the three set functions is decreasing a n d a t t a i n s its m i n i m u m at A = X. Moreover,
+ co >~u(A)>~v(A)>~w(A)> - oo.
I f the kernel satisfies F r o s t m a n ' s m a x i m u m principle, u ( A ) = v ( A ) for every set A . I t is well known, a n d will be shown in w 2.4, t h a t v ( A ) = w ( A ) for e v e r y set A if k is symmetric.
To each of t h e functions u, v, w corresponds an " e x t e r i o r " set f u n c t i o n defined as follows for a r b i t r a r y A c X :
u* (A) = sup u (G); v* (A) = sup v (G); w* (A) = sup w (G), (3) where G ~ ranges over the class of all open sets containing A. The relations u* ( A ) =
= u ( A ) , etc., hold for a n y open set A. I t will be shown presently t h a t t h e y hold likewise if A is c o m p a c t ( L e m m a 2.3.4).
The sets N c X such t h a t w (N) = + co or w* (N) = + co p l a y an i m p o r t a n t role as negligible sets. Observe t h a t each of these two classes of negligible sets remains u n c h a n g e d if the kernel k is replaced b y k + c for some c o n s t a n t c. I n t h e s t u d y of the two t y p e s of negligible sets it suffices, therefore, to consider case I (k~>0). A proposition involving a variable point x 6 A (where A denotes a given subset of X) is said to subsist nearly everywhere (n.e.) in A if w(~V)= + ~ , N being the set of points of A for which the proposition fails t o hold. Similarly, the proposition is said to hold quasi-everywhere (q.e.) in A if w* ( N ) = + ~ . T h e following l e m m a is easily verified, for instance in t h e succession (i) ~ (ii) ~ (iii) ~ (iv) ~ (v) ~ (i):
154 B E N T F U G L E D E
LEMMA 2.3.1. Let N c X . The /ollowing /ive conditions are equivalent:
(i) w ( N ) = + ~ .
(ii) w ( K ) = + oo for e v e r y c o m p a c t set K e N .
(iii) It. ( N ) = 0 for e v e r y positive m e a s u r e /z of finite e n e r g y on X.
(iv) /~ = 0 is the only positive m e a s u r e of finite energy c o n c e n t r a t e d on N . (v) It = 0 is the o n l y positive m e a s u r e of finite e n e r g y s u p p o r t e d b y some com- p a c t subset of N .
W e denote in t h e sequel b y 9~ the class of all sets which are m e a s u r a b l e with r e s p e c t to e v e r y m e a s u r e on X . I t follows f r o m condition (iii) of the a b o v e l e m m a t h a t the union o/ any denumerable /amily o/ sets o/ class 9~ with w = + cr is a set with w = + oo. T h e following l e m m a will be used later.
LEMMA 2.3.2. Let tz E ~ + , A c X , and 0 <~ t <~ + ~ . The/ollowing two propositions are equivalent:
(a) /c (x, It) >~ t for n e a r l y e v e r y x E A.
(a') k(v, it)>1 t . v ( X ) for e v e r y positive m e a s u r e v of finite e n e r g y a n d s u p p o r t e d b y some c o m p a c t subset of A.
I n p r o v i n g t h a t (a) implies (a'), we m a y suppose / c ( v , / u ) < ~ , i.e. k(x, i t ) i s v-integrable. W r i t i n g N = ( x E S ( v ) : I c ( x , / u ) < t } , we h a v e w ( N ) = + cr a n d hence v. ( N ) = 0. Since N is r - m e a s u r a b l e a n d contained in t h e c o m p a c t s u p p o r t of v, we conclude t h a t v* ( N ) - 0, a n d hence
k(v, i t ) = f k(x, # ) d v > ~ t . f d v = t . v ( X ) .
S 0') S (v)
N e x t , suppose (a) is n o t fulfilled, a n d write B = (x E A : k (x, ~) < t). T h e n w (/3) 4 § ~ , a n d hence t h e r e is, in view of L e m m a 2.3.1, a positive m e a s u r e v ~ 0 of finite e n e r g y s u p p o r t e d b y some c o m p a c t set K c B . Since k ( x , # ) < t in K , we o b t a i n k ( r , / ~ )
< t . v ( X ) in contradiction w i t h (a').
T H E O R E M 2.3. For any non-empty compact set K c X , each o/the three in/ima (1), with A = K , is an actual minimum. The minimizing measures constitute a compact sub-
class o/ ~+.
Proo/. Choose a function ] E C ~ which equals 1 in K . T h e n j u ( X ) = f / d # de- pends] continuously on /t E 7~/~. T h e subset of ~ + o v e r which /~ ranges in the ex- t r e m a l p r o b l e m s (1) is therefore closed (in t h e v a g u e topology). Being v a g u e l y bounded, this set is c o m p a c t (and n o n - e m p t y w h e n K is n o n - e m p t y ) . H e n c e the t h e o r e m fol-
O N T H E T H E O R Y O F P O T E N T I A L S I N L O C A L L Y C O M P A C T S P A C E S 155 lows f r o m L e m m a 2.2.1. W e d e n o t e t h e t h r e e classes of m i n i m i z i n g m e a s u r e s b y
~ , ~K, a n d ~ x , r e s p e c t i v e l y .
L E M ~ A 2.3.3. I / A denotes the union o/ an increasing sequence o/ sets A~ o/
class 9X, then
u ( A ) = l i m u ( A . ) ; v ( A ) = lira v(A~); w ( A ) = l i m w(A~).
Proo[. I t suffices t o consider case I (k>~0), a n d we m a y a s s u m e t h a t A is non- void. L e t /~ E ~ , # ( X ) = 1, a n d d e n o t e b y /,~ t h e t r a c e of # on t h e /~-measurable set A~. T h e n /,~ (X) = # ( A , ) - + ~u (A) = fl (X) = 1, a n d hence we m a y s u p p o s e all #~ 4: 0.
W r i t i n g ~ = # , / / , (A~), we h a v e ~ E ~$~+ a n d ~n (X) = 1. H e n c e An w (A,) < W ( ~ ) = W (/~.)/~ (A.) 2 <-< W (~)//~ (A.) ~.
L e t t i n g n --> ~ , we c o n c l u d e t h a t lira w (An) ~ W (#), which i m p l i e s t h e n o n - t r i v i a l p a r t of t h e l i m i t r e l a t i o n for w. T h e p r o o f is s i m i l a r for t h e f u n c t i o n s u a n d v.
Remark. F o r a decreasing sequence t h e c o r r e s p o n d i n g l i m i t r e l a t i o n w o u l d b e false in general. (1) I t does hold, however, i n t h e case of compact sets; a n d in t h a t case t h e sequence m a y e v e n be r e p l a c e d b y a n a r b i t r a r y d e c r e a s i n g filtering ]amily of c o m p a c t sets K . W r i t i n g K 0 = N K , we p r o p o s e t o e s t a b l i s h t h e following
K e ~
relations:
u (Ko) = l i m u (K); v (Ko) = lim v (K); w (Ko) = l i m w (K) (4) as K - > K 0 t h r o u g h ~ . (Of course, t h e l i m i t sign m a y be r e p l a c e d b y s u p r e m u m o v e r all K E ~ . ) Consider, e.g., t h e set f u n c t i o n w, a n d choose for e v e r y K E ~ s o m e m i n i - m i z i n g m e a s u r e #K E WK (i.e. jUK E ~ : , # g (X) = 1, W (/~K) = w (K). W e d i s r e g a r d t h e t r i v i a l case where s o m e K is e m p t y ) . Since t h e m e a s u r e s #K, K E ~ , belong t o t h e b o u n d e d (i.e. r e l a t i v e l y c o m p a c t ) class of all m e a s u r e s # 6 ~ + w i t h # ( X ) ~ 1, t h e r e exists a t l e a s t one c l u s t e r p o i n t /~0 of /~K as K - > K 0 t h r o u g h ~ . B y v i r t u e of t h e l o w e r s e m i - c o n t i n u i t y of W(/~) a n d t h e f a c t t h a t w is decreasing, we c o n c l u d e t h a t , as K - - > K 0 t h r o u g h ~ ,
W (~0) < lim w (K) < w (K0). (5)
C l e a r l y /~o is s u p p o r t e d b y e a c h K E ~ a n d h e n c e b y K o. I t will b e s h o w n p r e s e n t l y (1) As an example pertaining to the Newtonian kernel I x - y ] 1 in R 3, let An denote the open region between two concentric spheres, say Au = (x e Ra: 1 - n -1 < Ix ] < 1}. Then fl An is void, but w (An) = 1. n
11 - 603808 Acta mathematica. 103. Imprim6 le 22 juin 1960
156 BE~T FUGLEDE
t h a t #0 ( X ) = 1. T h e n i t follows f r o m (5) t h a t w ( K 0 ) = W (#0)- This implies, i n t u r n , t h e d e s i r e d r e l a t i o n (4) a n d , m o r e o v e r , t h a t /t o is a minimizing measure /or the set K0 .(1) To see t h a t / t 0 ( X ) = l , we choose a f u n c t i o n / E C ~ so t h a t / ( x ) = l i n s o m e o p e n set G ~ K o. N o w , G c o n t a i n s s o m e set H E ~ . ( 2 ) F o r e v e r y K E ~ such t h a t K ~ H we h a v e
f
a n d hence /to (X) = f / d / t o = lim f [d/tK= l.
L ~ M ~ A 2.3.4. For any compact set K, u * ( K ) = u ( K ) , v * ( K ) = v ( K ) , and w*(K)
= w ( K ) .
This follows f r o m t h e a b o v e r e m a r k a p p l i e d t o t h e f i l t e r i n g f a m i l y of all com- p a c t n e i g h b o u r h o o d s H of K . I t is well k n o w n t h a t t h e i n t e r s e c t i o n of t h i s f a m i l y is K . H e n c e t h e r e c o r r e s p o n d s t o e v e r y n u m b e r t < w ( K ) a c o m p a c t n e i g h b o u r h o o d H of K such t h a t w ( H ) > t . I f G d e n o t e s t h e i n t e r i o r of H , w e h a v e G ~ K , a n d h e n c e
w* (K) >~ w (G) >i w (H) > t.
C o n s e q u e n t l y , w* (K) >1 w (K), q.e.d.
This l e m m a , or t h e r e m a r k on w h i c h i t was b a s e d , a s s e r t s t h a t t h e d e c r e a s i n g set f u n c t i o n s u, v, a n d w a r e continuous from the outside w h e n c o n s i d e r e d on c o m p a c t sets. :Now, let 0 = 0 (t) d e n o t e a decreasing f u n c t i o n of t h e r e a l v a r i a b l e t, m a p p i n g t h e i n t e r v a l w (X)~< t ~ + co (in case of t h e set f u n c t i o n w) in a c o n t i n u o u s w a y i n t o t h e e x t e n d e d r e a l line. T h e set f u n c t i o n ~ ( K ) = 0 (w(K)), c o n s i d e r e d on t h e class of (1) The p a r t of this remark which concerns the minimizing measures can be formulated in a slightly stronger way in which the arbitrary choice of a minimizing measure for each set K E ~ is avoided. We associate with every set H E ~ t h e "section"
$ ( ~ ) = U ~ K ( H e ~ , K e ~ )
K C H
/onsisting of all minimizing measures for all subsets K c H, K E ~. These sections S (H) generate a ilter (0 on a relatively compact subset of ~ + (because fl (X) = 1), and the proof above shows t h a t t h e vague adherence of 9 is contained in ~ K , .
(2) I n fact, t h e decreasing filtering family formed b y t h e compact sets K fl C G, K E ~, has a void intersection. According to the "finite intersection principle", there is a finite family ( K t ) i = l , n
Ki E ~, such t h a t the intersection
n n
( N K0 N C a = n (K~ (~ [J G)
i = l ~ = 1
is void. Since ~ is filtering, there is a set H E ~ such t h a t H c N Ki, and hence H ~ G.
1 = 1
O N T H E T H E O R Y O F P O T E N T I A L S I N L O C A L L Y C O M P A C T S P A C E S 157 all compact sets, is then a
capacity
in t h e sense of Choquet [14], w 15, (i.e. an in- creasing set function which is continuous from the outside). According to (2), resp. (3),y , ( A ) = O ( w ( A ) )
andy*(A)=O(w*(A))
are the corresponding interior, resp. exterior, capacities. A set A is called
capacitable
if y . (A) = y* (A). F o r such a set we write simply )~(A) instead ofy . ( A )
or y*(A).Similarly, one m a y study the capacities c~ (K) = 0 (u (K)) and/3 (K) = 0 (v (K)). As shown above, open sets and compact sets are capacitable. Results concerning capaeitability of more general sets can be obtained b y application of Choquet's theory, a t least under suitable assumptions concerning the kernel k and the space X, el. M. Kishi [19] in the case of a kernel k~> 0 satisfying lq'rostman's m a x i m u m principle. I n w 4 of the present study we obtain somewhat stronger results in case of a consistent kernel.
We shall now consider the following choice of the function 0:
1
O ( t ) = t - a '
where a denotes a real constant
<~w(X)
(resp.u ( X )
orv(X)).
I n particular, a n y n u m b e r a~< inf k (x, y) can be used. The most i m p o r t a n t case is a = 0, which leads to the Wiener capacity, cf. w 2.5.LEMMA 2.3.5.
I/ a denotes a constant such that k(x, y)>~a, then the interior ca- pacity ( w ( A ) - a ) -1 is countably subadditive on sets o] class 9~, and the exterior capacity ( w * ( A ) - a ) -1 is countably subadditive on arbitrary sets. Similarly with w replaced by
u o r v .
Pro@
The kernel ] c ' = / c - a is positive, and the corresponding function w' equals w - a . Hence the capacity1/w'
equals the capacity in question. I t suffices, therefore, to prove the l e m m a for apositive kernel k
(with a = 0). Consider a sequence of sets An with the union A. Our task is to prove, first, t h a tw(A) -1<~ Z w(An) -1
provided AnEg~. (6)n
Without loss of generality we m a y assume t h a t the sets An are m u t u a l l y disjoint and t h a t A is not void. F o r a n y positive measure /z with compact support contained in A and with # ( X ) = I , we denote the trace of # upon
An
b y /z~ and write 2~=#~//z(A~) for such values of n t h a t # ( A ~ ) 4 0 . Neglecting the remaining values of n, if any, we have
1 5 8 B E N T F U G L E D E
w (A.) < k (~., ~.) = ~ (z., ~ ) / ~ (A~) ~, and hence, by application of Cauchy's inequality,
w (A~) -~ >1 ~ ~ (A~)~/k ( ~ , ~n) >1 (~ ~ (A~)iV5 k ( ~ , m)"
n n i t
The resulting inequality holds
a /ortiori
if the summations are extended overall
in- dices n. Note t h a t ~ / ~ ( A ~ ) = # ( A ) = / ~ ( X ) = I because the sets A n are /~-measurable and mutually disjoint. Hence (6) will follow if we can show t h a t~ ( ~ , ~ ) < ~ ( ~ , ~)-
n
This inequality is easily derived from the corresponding inequality in which the kernel ]c is replaced by an arbitrary function
/E C~ (X•
with / < k; and in that case we m a y apply (2), w 1.1 (with /% replaced by # | and A byA,~•
n n A n X A n
the sets
A~•
being mutually disjoint. Having thus established (6), we infer from the definition (3) of w* t h a tl/w*
is countably subadditive on arbitrary sets.--The corresponding assertions concerning the capacities1/u
and1Iv
m a y be verified simi- larly, but it is considerably simpler to make use of the following characterizations of1/u
andl/v,
valid ifu(X)>~O,
resp.v(X)>~O;
thus in particular if the kernel ]c is positive (]c ~> 0) or pseudo-positive (w (X) >~ 0):1 / u (A) = sup 4. (A) (4 e ~ ; k (x, 4) ~ 1 everywhere)
1 / v ( A ) = s u p
2.(A)
( 2 E ~ ; k(x, 2)~<1 for x E S ( ~ ) ) . (7) COROLLARY. Let k denote an arbitrary kernel (~> 0 unless X is compact), and let N denote the union of a sequence of sets -h~ c X. If N n E ~ and w (N~) = + 0o, then w ( N ) = + ~ . If w*(N~)= + ~ , thenw*(N)= + ~ .
Likewisew ( A U N ) = w ( A )
if w ( N ) = + ~ and A, NEg~;w*(A U N)=w*(A)
ifw*(N)= + ~ .
Similar statements apply to the set functions u and v.Remark.
In Lemma 2.3.5 (and its corollary above) the assumption t h a t the sets An (resp. Nn) be of class i~ (in case of the functionsu, v, w)
m a y be replaced b y the slightly weaker hypothesis t h a t A~ = A~ N B, where A~ E ~, whereas B is arbitrary.Writing A ' = U A~, we have then
A =A' N B.
SinceS ( # ) c A ~ B ,
the set B is ,u-meas-n
O1~ T H E T H E O R Y O F P O T E N T I A L S I N L O C A L L Y COMPACT S P A C E S 159 u r a b l e , a n d so are, therefore, t h e sets An. N o t e t h a t t h e r e s t r i c t i o n t o p o s i t i v e k e r n e l s is i n d i s p e n s a b l e in L e m m a 2.3.5 (cf. E x . 2, w 8.3).
2.4. The case o/ a symmetric kernel. W h e n t h e k e r n e l k is s y m m e t r i c , t h e t w o set f u n c t i o n s v a n d w a r e i d e n t i c a l , a n d so a r e v* a n d w*. I t suffices t o p r o v e t h a t v ( K ) = w ( K ) for e v e r y compact set K . Since we k n o w t h a t v~>w, we m a y a s s u m e t h a t w ( K ) < + oo. W e p r o p o s e t o v e r i f y , m o r e o v e r , t h a t t h e t w o m i n i m u m p r o b l e m s defining v ( K ) a n d w ( K ) (ef. T h e o r e m 2.3):
v (K) = m i n V (/t); w (K) = m i n W (/t),
(where in b o t h cases /t E ~ : a n d / t ( K ) = 1) h a v e p r e c i s e l y t h e same solutions, i.e.
~ K = ~ g . I n t h e t h e o r e m b e l o w we p r o v e t h a t /t E 7~K' i m p l i e s V(/t)~<w (K) ( p r o p e r t y (b)). Since w (K) ~< v (K), we infer t h a t /t E ~K a n d t h a t , a c t u a l l y , w (K) = v (K). Con- v e r s e l y , /t E 10K i m p l i e s k ( / t , / t ) ~< V (/t) = v (K) = w (K), w h i c h shows t h a t # E WK. Con- s e q u e n t l y , ~K = 7~qK. T h e s o l u t i o n s /t E WK of t h e second (and hence of t h e first) m i n i - m u m p r o b l e m a b o v e will be called capacitary distributions o/ unit mass on K .
THEOREM 2.4. Let k denote a symmetric kernel, and K a compact set such that w (K) < § oo. The potential o/ any capacitary distribution # o/ unit mass on K has the /ollowing properties:
(a) k (x, #) >~ w (K) n e a r l y e v e r y w h e r e i n K .
(b) k (x, #)~<w (K) e v e r y w h e r e in t h e s u p p o r t of /t.
(c) k (x, # ) = w ( K ) / t - a l m o s t e v e r y w h e r e in X .
Proo/. W e b e g i n b y e s t a b l i s h i n g (a) in t h e e q u i v a l e n t i n t e g r a t e d f o r m (cf.
L e m m a 2.3.2):
(a') k ( ~ , / t ) > ~ w ( K ) . ~ ( X ) for e v e r y v E T ~ w i t h k(v, v ) < § oo.(1)
I t suffices t o consider t h e case w h e r e k(v, # ) < + 0 0 a n d v ( X ) = l . T h e n a # + b v is a
" c o m p e t i n g " m e a s u r e for a n y choice of c o n s t a n t s a ~ 0 , b>~0, w i t h a § 1. H e n c e its e n e r g y
a2k(/t, /t)+ 2 a b . k ( v , # ) + b 2 k ( ~ , r)
a t t a i n s its m i n i m u m a t a = l , b = 0 . T h i s i m p l i e s (a') w h e n i t is o b s e r v e d t h a t (1) The assumption k (v, v) < + c~ is essential, not only for the proof, but~ for the validity of the result. Otherwise one could take v = ~x ( = the mass § 1 placed at an arbitrary point x E K) and con- clude that k (x, tt) >i w (K) everywhere in K. This would be false even in case of the •ewtonian kernel (unless the unbounded component of C K is regular for Dirichlet's problem).