A R K I V
FOR
M A T E M A T I K B a n d 4 n r 9Communicated 9 September 1959 b y OTTO FROST~IAN and LENNART CARLESObr
One-parameter semigroups of subsets of a real linear space
B y HANS R.~DSTRbM
C O N T E N T S
w 1. I n t r o d u c t i o n . . . . 87
w 2. R e m a r k s a n d e x a m p l e s . . . 88
w 3. A u x i l i a r y m a t e r i a l . . . 91
w 4. O n e - p a r a m e t e r s e m i g r o u p s of c o n v e x sets . . . 94
w 5. O n e - p a r a m e t e r s e m i g r o u p s of c o m p a c t sets, s t r u c t u r e t h e o r e m . . . . 96
w 1. Introduction
The main objective of this paper is to establish the structure theorem in w 5.
If K and M are subsets of a linear space we let K + M denote the set of all sums k + m , where k E K and mEM. If K consists of only one point k, we write k + M instead of {k} + M. This operation of addition is obviously commutative and associa- tive. Thus a family of subsets forms a commutative semigroup under addition if it is closed under this 6peration. Such a semigroup is called a one-parameter semigroup if there exists a one-to-one application: 6-->A (r of the set of positive real numbers onto the semigroup, satisfying:
A ((31 -~ (32) = A (~1) "~- A (62)"
I n his paper (2) Gleason introduced the concept of a one-parameter semigroup of subsets of an arbitrary toplogical group. I n m y thesis (3, henceforth referred to as CN) I obtained results concerning the structure of such semigroups in Lie groups.
I n particular, it follows from results in CN that in a finite-dimensional linear space every one-parameter semigroup A (d) of closed sets which come arbitrarily close to the origin as (3-->0 is of the form A (~) = (3.M where M is a convex, compact set and 6 M denotes the set of all ~m with m EM. This is of course a consequence of the much more general results in w 5 below.
I t should be observed that the definition of one-parameter semigroups given above is neither equivalent to the one in CN nor to Gleason's although there are no very essential differences.
We remark that:
1 ~ I n CN a one-parameter semigroup is the mapping d-->A(5), and not a family of sets. Thus in CN A (~) could be constant which is not possible with the present definition. (We require 5-->A ((~) to be one-to-one.)
87
H. Rs One-parameter semigroups
2 =' I n CN t h e m a p p i n g is d e f i n e d for d ~ 0 a n d n o t o n l y f o r / i > 0.
3 ~ I n CN t h e sets A (8) are a s s u m e d t o be closed for all small 8.
4 ~ I n CN t h e m a p p i n g is a s s u m e d t o be c o n t i n u o u s w i t h r e s p e c t t o H a u s d o r f f ' s t o p o l o g y for sets (see below, s e c t i o n 3.1).
If t h e s e m i g r o u p is p a r a m e t r i z e d in t w o w a y s w i t h p a r a m e t e r s 5 a n d e, t h e n t h e I e e x i s t s a one-to-one m a p p i n g , [, of t h e set of p o s i t i v e reals o n t o itself satisfying
e = [(b) a n d ](6: + d2) = ](81) + [(~2)-
I t is well k n o w n that, t h e only p o s i t i v e solutions of this f u n c t i o n a l e q u a t i o n are of t h e form /(5) - c o n s t . 8. Therefore e v e r y possible r e p a r a m e t r i z a t i o n of a given s e m i g r o u p consists s i m p l y in a change of u n i t for t h e p a r a m e t e r .
w 2. Remarks and examples 2.1, W e d e n o t e b y 0 M t h e set of all 8 m where m E M .
E x a m p l e 1. I f M is c o n v e x i t is e a s y t o p r o v e t h a t (81 + 82) M = 81M + 82 M- Thus, for a c o n v e x set M , which is n o t a cone, t h e sets 8 M c o n s t i t u t e a o n e - p a r a m e t e r semi- g r o u p w i t h 5 as p a r a m e t e r . (We m u s t require t h a t M n o t be a cone in o r d e r t o g u a r a n - tee t h a t t h e m a p p i n g 8---~8M be one-to-one.)
Conversely, s u p p o s e t h a t M is a set such t h a t t h e sets 8 M f o r m a o n e - p a r a m e t e r s e m i g r o u p with 5 as p a r a m e t e r . T h e n M is conve2. N a m e l y , let x a n d y be e l e m e n t s of M . Then 2 x E 2 M a n d (1 - ) , ) y C ( 1 - ) , ) M for 0 < 2 < 1. I t follows t h a t ),x + (1 - X ) y C "
E ) , M + (1 - 2 ) M = M .
2.2. De/inition: A one-parameter semigroup A (8) is called linear i / t h e r e is a set M such that
A (8) = d M for all 6 > O.
W e h a v e seen t h a t a l i n e a r s e m i g r o u p consists of c o n v e x sets. The converse is n o t true, n o t even in t h e case of a o n e - d i m e n s i o n a l space as t h e n e x t e x a m p l e proves.
E x a m p l e 2. L e t [ (d) be a r e a l v a l u e d f u n c t i o n of t h e ~ p o s i t i v e r e a l v a r i a b l e 8 a n d s a t i s f y i n g t h e f u n c t i o n a l e q u a t i o n
/ (81 + 52) = ] (61) -~ / (82).
P u t t i n g A (d) = {[(8)} for each 8 we o b t a i n a o n e - p a r a m e t e r s e m i g r o u p of p o i n t s on t h e real line. I t is k n o w n t h a t u n d e r a l m o s t a n y (even v e r y mild) r e g u l a r i t y c o n d i t i o n t h e f u n c t i o n /(5) m u s t be of t h e f o r m c o n s t . 8 a n d c o n s e q u e n t l y A (8) linear. On t h e o t h e r h a n d t h e r e also exist p a t h o l o g i c a l solutions of t h e e q u a t i o n . T h u s a s e m i g r o u p of p o i n t s n e e d n o t be linear.
2.3. The r e s u l t in w 5 below shows t h a t for o n e - p a r a m e t e r s e m i g r o u p s of c o m p a c t sets in a l o c a l l y c o n v e x space, t h e t w o e x a m p l e s g i v e n cover all possibilities. W e shall see t h a t if A (5) is such a semigroup, i t can be e x p r e s s e d in t h e form A (8) = / (8) + 8- M , where [ (d) is a s e m i g r o u p of p o i n t s a n d M a c o m p a c t c o n v e x set. I f t h e c o m p a c t n e s s c o n d i t i o n is n o t m a d e t h e conclusion A (d) = [(8) + 8 M w i t h c o n v e x M is no longer
~s
ARKIV F()n MATEMATIK. B d 4 nr 9
Example 3. On the real line, let A (6) be the set of all rational numbers r with 0 < r < 6 .
T h e n the sets A (6) form a o n e - p a r a m e t e r semigroup of non-convex sets.
However, in this example the closure, A (6), of each set in the semigroup is convex.
Examples 5, 6 a n d 7 below show, a m o n g other things, t h a t this is n o t always the case.
2.4. We shall also be s o m e w h a t interested in finding conditions to ensure t h a t A (~) or A (5) is linear. If A (6) is given a n d r (6) is a o n e - p a r a m e t e r semigroup of points, then r + A (6) is a one-parameter semigroup. Can r be chosen so t h a t r +
+ A (5) is linear? Our next example shows t h a t this need not. be possible even if the sets A (6) are convex.
Example 4. L e t /(6) be a pathological function in example 2. I n the xy-plane, consider the set A (6) of all points (x, y) with 0 < x < 6 together with the points (0, 0) a n d (5,/(6)). T h e n the sets A(6) form a non-linear, one-parameter semigroup of convex sets such t h a t r + A (5) is n o t linear for a n y choice of r
2.5. The following s t a t e m e n t is an application of Theorem 5.6 of CN (p. 133) to t h e case of a linear space: Let a translation i n v a r i a n t metric be given on a linear space a n d denote the closed sphere of radius 6 a r o u n d the origin b y S(d). A necessary a n d sufficient condition t h a t S (6) forms a o n e - p a r a m e t e r semigroup is t h a t the metric be convex. (See CN p. 133.) A well k n o w n case w h e n this occurs is when the metric is defined b y a norm. I t can, however, occur also in other cases.
Example 5. Consider for 0 < p ~< 1 the space L p of functions [(t) (with the usual
1
identifications) defined for 0 ~<t ~< 1 a n d such t h a t II/ll
-fl/(t) l "dt
< ~ ' ( O b s e r v e !0 1
Unusual notation. I f p ~ > l the n o r m is defined b y ]]/IP = f l / ( t ) l pdt. Our ]]/ll is
0
not u n o r m for p < 1 since it is n o t homogeneous.) I f t t / - g II is t a k e n to m e a n the distance between / and g this makes the space a complete metric space which is not locally convex for p < 1 b u t locally b o u n d e d (see T y c h o n o f f 4, or Bourbaki 1 p. 224 ex 17). The triangle inequality follows from the e l e m e n t a r y inequality
la+bl'<lal'+lbl"
in which for p < 1 the sign of equality holds if and only if either a or b is zero. Thus the triangle inequality is strict unless t h e t w o functions involved are different from zero on disjoint sets in which ease equality holds. N o w t h e triangle inequality:
Iit +gll < II/ll + Ilglt
is obviously equivalent to the following inequality for the spheres:
W e obtain equality if a n d only if to a n y h with II h II ~< 61 + 62 it is possible to find [ a n d g with / + g -- h, / ~< 6~ a n d II g II ~< 62. I t is obviously sufficient to prove t h a t g i v e n h with I h l l = 6 1 + d ~ , there exist / a n d g with [ + g = h a n d 11/11=51 and II g II = 62. This is done in t h e following way. L e t 2 be a n u m b e r such t h a t
H, R~DSTR6M,
One-parameter semigraups
f lh(t)l= dt=81
0 1
which exists since 0 < (~1 < 81 ~- 82 = f I h l ' ~ 0 P u t
h (t) for f (t) = 0 for
0 < t < ~ 2~<t~<l f 0 for 0 ~ < t < 2 g ( t ) =
h(t) for 2-<<t~<l
Since the sets where ] a n d g are different from zero are disjoint t h e result follows.
Thus the spheres form a one-parameter semigroup a n d t h e metric is convex. Since the spheres are closed and it is known t h a t t h e y are bounded, this is also an example t h a t a one-parameter semigroup of closed bounded sets need not consist o/convex sets.
2.6. The spaces L v for p < l are n o t locally convex and for p = 1 the semigroup of spheres in example 5 does in fact consist of convex sets. Thus we have n o t y e t obtained an example of a pathological ( = consisting of non-convex sets) semigroup of closed b o u n d e d sets in a locally convex space. E x a m p l e 6, however, shows t h a t such an example exists in L 1.
Example 6. I n example 5 let p = 1 and p u t A (8) = the set of those functions ] which satisfy
1 1 ~ j f , z t = 8
o
2 ~ f / > O
3 ~ The values of ] are integers almost everywhere.
Then I]/II = 8 so t h a t A (8) c S(8). Thus A (8) is b o u n d e d a n d it is obviously closed.
N o w let ]fiA (81) and gEA (82). Then f + g is clearly in A (81 + 82) which shows t h a t A (81) + A (82) c A (81 + 82). The reversed inequality is obtained b y considering an element h in A (81 + 82) and applying the decomposition of example 5. This yields ] e A (81) a n d g e A (82).
N o w A (8) is n o t a semigroup of convex sets. We prove t h a t A (89 is n o t convex.
L e t fl a n d / 2 be defined as follows
1 if O~x~< 89 ] l ( x ) = 0 if 8 9 ] 2 ( x ) = { 0 if 0 < x < 8 9
1 if 8 9
Thus ]1 and ]2 are elements of A (89 b u t (fl + f2)/2 = 89 for all z a n d is therefore n o t integervalued almost everywhere and so it is n o t an element of A (89
90
ARKIV F{JR MATEMATIK. Bd 4 nr 9 2.7~
Example 7. Let A ((~) be the set of all integervalued functions in L 1 (0, 1) of norm ~< ~$.
Then A ((~) is closed and bounded and A (89 contains the functions ]1 and ]2 of example 6, but it does not contain the function (]1 + [2)/2. Thus the function (]1 +/2)/2 has a po- sitive distance to the closed set A (89 and therefore there is a positive number k so small t h a t A(89 + kS (89 is not convex. (S((~) denotes the sphere of radius c$ in the norm.) Hence the semigroup $1((~ ) = A (~)+ kS((~) does not consist of convex sets.
On the other hand all these sets are symmetric neighborhoods of the origin and again b y Theorem 5.6 of CN they are the spheres around the origin in a translation invariant convex metric for L 1. This shows therefore that such spheres need not be convex even in a locally convex space.
w 3. A u x i l i a r y m a t e r i a l 3.1. Hausdor//'s topology /or sets.
The following is a well known method to define a topology on the family, F, of all subsets of a linear space. Let U be a neighborhood of the origin and A a given set, the neighborhoods in F of which we want to define. Let U' (A) be the subset of F consisting of those sets B in the space for which the two inequalities hold:
A c B + U B c A + U .
If U varies in the set of all neighborhoods of the origin, the corresponding sets U' (A) run through a fundamental system of neighborhoods of A in a topology for F. This will be called HawsdorH's topology.
In general (.i.e. if the space has more than one point) Hausdorff's topology does not satisfy Hausdorff's separation axiom, T~, for it is clear that if A is not closed the sets (or points in F) A and ~ are two different elements of all sets U'(A) or U'(~). On the other hand, the restriction o] Hausdor[f s topology to the family of closed subsets of the linear space satisfies T 2 and so does the restriction to the open convex subsets. The first part of this proposition is standard and the second is an immediate consequence of the following theorem in the theory of convex sets: If A is open and convex then A = I n t ] .
Although limits are not unique in Hausdorff's topology we can thus nevertheless pick at most one limit among the closed sets and if there exists an open convex limit it is also unique.
3.2. The convex hull.
We denote the convex hull of a set A b y H A . The closed convex hull will be denoted by ]qA. I n this section we prove the formula:
H ( A + B) = H A + H B
which is important in the sequel and, rather surprisingly, does not seem to be ex- plicitly mentioned in the literature on convex sets.
Although the formula follows rather easily from the barycentric representation of the points in the convex hull, we prefer to deduce it as a special case from the following consideration which is much to general for our immediate purpose but has also other applications.
H. R.~DSTROM,
One-parameter semigroups
B y a hull operation we m e a n a n o p e r a t i o n , T, which is d e f i n e d on a f a m i l y of s u b s e t s of a given space, t a k e s its v a l u e s in t h e s a m e f a m i l y a n d w h i c h satisfies
1 ~ T A r A 2 ~ T2A = T A
3 ~ A c B implies T A c T B .
I f we call a n y set A s a t i s f y i n g T A = A a hull, i t is i m m e d i a t e t h a t for a n a r b i t r a r y set A t h e set T A is t h e i n t e r s e c t i o n of all hulls c o n t a i n i n g A. W e r e m a r k also t h a t if T is n o t defined for all s u b s e t s of t h e space, we can e x t e n d i t b y defining T A as t h e i n t e r s e c t i o n of all hulls c o n t a i n i n g A if t h e r e is such a hull a n d as t h e e n t i r e space otherwise. N o w s u p p o s e t h a t t h e space is a l i n e a r space a n d t h a t our hull o p e r a t i o n is translation invariant, t h a t is
Then we have
4 ~ x + T A = T ( x + A ) for all x in t h e space.
T A + T B ~ T ( A + B) and equality holds if and only if the left member is a hull.
Proof. L e t a E A . T h e n b y 4 ~ a n d 3 ~
a + T B = T (a + B) c T (A + B).
I t follows t h a t
(1) A + T B c T ( A + B).
S u b s t i t u t i n g T A for A we o b t a i n
T A + T B c T ( T A + B ) a n d b y use of (1)
T A + T B c T T ( A + B) = T ( A + B), which p r o v e s t h e first p a r t of t h e p r o p o s i t i o n .
N o w observe t h a t f r o m A c T A a n d B ~ T B follows A + B c T A + T B .
T h u s T ( A + B) c T ( T A + T B ) c T 2 ( A + B ) = T ( A + B) so t h a t T ( T A + T B ) = T (A + B). I n t h i s e q u a t i o n t h e left m e m b e r e q u a l s T A + T B if a n d o n l y if t h i s set is a hull, w h i c h p r o v e s t h e s e c o n d p a r t of t h e p r o p o s i t i o n .
As special cases we t a k e T = H a n d o b t a i n t h e f o r m u l a which we set o u t t o prove.
O t h e r specializations are o b t a i n e d b y l e t t i n g T A d e n o t e t h e l i n e a r v a r i e t y s p a n n e d b y A or t h e a d d i t i v e v a r i e t y g e n e r a t e d b y A. I n t h e s e cases we h a v e T A + T B = T (A + B). A n e x a m p l e where e q u a l i t y does n o t a l w a y s h o l d is f u r n i s h e d b y choosing T t o be t h e o p e r a t i o n of closure: T A = . ~ . T h u s A + / ~ c A + B a n d e q u a l i t y holds if a n d o n l y if A + / ~ is a closed set. I n p a r t i c u l a r e q u a l i t y holds if A or B has c o m p a c t closure.
3.3. The support function.
L e t T be a l i n e a r s y s t e m . B y T * we d e n o t e t h e a l g e b r a i c a d j o i n t of T, t h a t is t h e linear s y s t e m consisting of all l i n e a r f u n c t i o n a l s on T. L e t L b e s u c h a f u n c t i o n a l
ARKIV FOR MATEMATIK. B d 4 n r 9 a n d K a set i n T. B y L o K we d e n o t e the s u p r e m u m of L on K. This s u p r e m u m m a y of course a s s u m e t h e value + ~ , b u t if K is n o n - e m p t y it does n o t assume t h e v a l u e
- co. I f K a n d M are two n o n - e m p t y sets we have
3.31. L o ( K + M ) = L 9 + L o M ,
where i n t h e case w h e n t h e v a l u e + ~ occurs the obvious c o n v e n t i o n s "oo + real = ~ "
a n d "c~ + ~ = cr should be made. If t h e set K is k e p t fixed a n d the f u n c t i o n a l L varies i n T * the m a p p i n g L - + L o K is a (real- or + c~-valued) f u n c t i o n o n T * which is called t h e s u p p o r t f u n c t i o n SK of K . F o r m u l a 3.31 above shows t h a t
3.32. SK+M = S~ + S~.
I t is easy to set t h a t t h e s u p p o r t f u n c t i o n is positively homogeneous:
SK(~L) =2SK(L) if ~ 7> 0, a n d subadditive:
SK (L1 + L2) <- SKL1 + SKL2.
I t is therefore a convex f u n c t i o n .
F o r a o n e - p o i n t set K t h e s u p p o r t f u n c t i o n is a linear a n d everywhere finite- v a l u e d f u n c t i o n o n T * , i.e. SK is a n e l e m e n t of T**. The m a p p i n g of T i n t o T * * defined i n this way:
a-->Z{a};
is called t h e canonical e m b e d d i n g of T i n T * * . I n t h e following considerations we shall assume t h a t T is e m b e d d e d i n T * * i n this way.If K c T is a t r a n s l a t e of M c T , s a y K ~ M ~- a, t h e n 3.32 gives
3.33. SK = SM +
S{~}
where the second t e r m i n t h e right m e m b e r is a l i n e a r f u n c t i o n a l o n T*. Conversely, suppose we know t h a t SK = SM + (line,~r f u n c t i o n a l o n T*). Does it follow t h a t K is a t r a n s l a t e of M? The answer is yes if b o t h sets are convex a n d compact in a locally convex, H a u s d o r f f topology for T. I n fact
3.34. Let a locally convex, Hausdor// topology be given on T. Then there exists a projection P o / T * * onto a subspace containing T such that i/
1 ~ K and M are compact, convex subsets o/ T and s E T**,
2 ~ SK(L) = SM(L) + s ( L ) /or those /unctionals L E T * which are continuous on T, then K = M + Ps.
Proo[. A n y linear f u n c t i o n a l L o n T has a n a t u r a l e x t e n s i o n to T * * , for given s E T * * defined b y L (s) = s (L). W e a s s u m e henceforth t h a t t h e elements of T * are e x t e n d e d i n this way.
Now we consider t h e weak topology o n T * * defined b y tile set T" of f u n c t i o n a l s L which are c o n t i n u o u s o n T. This topology need n o t be H a u s d o r f f b u t t h e topology which i t induces on t h e s u b s e t T of T * * is Hausdorff since i t is t h e w e a k e n i n g of t h e original topology on T. Therefore K a n d M are c o m p a c t also i n t h e n e w topology.
T h e closure of t h e origin i n T * * is t h e i n t e r s e c t i o n R of all nullspaces of f u n c t i o n a l s i n T ' .
W e define P i n t h e following way. L e t T 1 be a s u p p l e m e n t of T + R i n T**.
T h u s T * * = T + R + T 1 a n d this is a decomposition of T * * i n a direct sum. We define P to be the p r o j e c t i o n o n T + T 1.
H. R~kDSTROM,
One-parameter semigroups
T r
:Now t h e r e l a t i o n SK = S M A7 8 s a y s t h a t for each L E : L o K = L o M + L(s) = L o ( M + s ) .
T h u s a n y closed halfspace in T * * which contains one of t h e sets K or M + s also contains t h e o t h e r set. This m e a n s t h a t t h e closed convex hulls of t h e sets coincide.
Since t h e sets are c o n v e x i t follows t h a t
(1) K = M + s .
Since R is t h e closure of t h e origin in T * * t h e set K" will c o n t a i n K + R. On t h e o t h e r h a n d t h i s set, being t h e s u m of a c o m p a c t set a n d a closed set, is closed, so t h a t we h a v e K = K + R. The s a m e a r g u m e n t applies t o M . S u b s t i t u t i n g in (1) we o b t a i n
(2) K + R = M + R +s.
W e now use t h e f a c t t h a t P is a linear t r a n s f o r m a t i o n so t h a t for a n y two sets M a n d N t h e f o r m u l a P ( M + N) = P (M) + P (N) is valid. A p p l y i n g P t o b o t h m e m b e r s of (2) a n d observing t h a t P ( K ) = K, P ( M ) = M a n d P ( R ) := 0 we o b t a i n
K = M + P s which was to be p r o v e d .
I n c i d e n t a l l y , it follows t h a t Ps E T so t h a t s E T + R. This shows t h a t c o n d i t i o n 2 ~ is a r a t h e r s t r o n g r e s t r i c t i o n on s. F o r in general, of course, T + R is v e r y far from being all of T * * .
w 4. One-parameter semigroups of convex sets
4.1. W e suppose in t h i s p a r a g r a p h t h a t the sets A ((~) are convex and ]orm a one.
parameter semigroup. Then/or arbitrary ~ > 0 and rational 5 > 0:
A ( ~ ) = ~ A (~).
Proo/. W e h a v e A ( A ) = A ( ~ ) + A ( ~ ) + . . - + A ( ~ ) , ( n t e r m s ) , which i s =
= n A ( ~ ) since A ( ~ ) i s convex. T h u s A ( ~ ) =I-A(A)'n B y a c h a n g e of n o t a - t i o n we h a v e also A ( m ) , ) = m a ( A ) . C o m b i n i n g these t w o f o r m u l a s we o b t a i n
4.2. E x a m p l e 2 (section 2.2) shows t h a t t h e a b o v e p r o p o s i t i o n need n o t be t r u e for all ~ > 0, i.e. a o n e - p a r a m e t e r s e m i g r o u p of c o n v e x sets n e e d n o t be linear. I / w e make the/ollowing topological assumptions, however, the linearity /ollows a l m o s t t r i v i a l l y .
1. The convex sets A (5) are bounded and either all closed or all open.
2. The mapping A is continuous.
I t is e a s y to see t h a t for a b o u n d e d set K t h e sets 5 K v a r y c o n t i n u o u s l y w i t h 8.
(Not t r u e for all u n b o u n d e d sets. C o u n t e r e x a m p l e see CN p. 106.) F r o m 4.1 we h a v e 94
A R K I V F O R M A T E M A T I K . Bd 4 n r 9 A (6) = 6 A (1) for rational 5 > 0.
Since b o t h members v a r y continuously with 6 the result would follow if Hausdorff limits were unique. N o w the result follows from proposition 3.1. E x a m p l e 2 shows t h a t the assumption 1 alone does n o t imply linearity. E x a m p l e 4 shows t h a t neither does assumption 2.
4.3. The following result is basic in the proof of the m a i n theorem in w 5.
Let A (5) be a one.parameter semigroup of compact convex sets in a Hausdor/f, locally convex linear space, and let a E A (1) be given. Then there exists a function f defined for all 6 > 0 taking (point) values in the space and satisfying
1. f(6: + 62) =/(61) +/(62) 2. /(6)~A(6)
3. f(1) = a.
Proof. L e t the space be T and its topological adjoint T'. F o r each 6 > 0 we denote b y ~(6, L) the restriction to T ' of the support function of A (6). Then ~0 is a realvalued function. F r o m the semigroup p r o p e r t y of A (5) it follows t h a t ~0 is additive as a func- tion of 6 for each L. Since ~0 is a support function it is subadditive as a function of L for each 6 > 0.
N o w let L 1 and L~ be two elements of T'. B y the subadditivity in L of ~0 (6, L) we have
~0 (6, i l ) + ~0 (5, L2) - ~ (6, L 1 § L2) ~ 0.
The left m e m b e r of this inequality is an additive real-valued function %0 (6) for 6 > 0.
I t is wellknown t h a t such a function can be non-negative only if ~p(6) = 6%o(1). Thus
~0(6) -6%0(1) = 0 or in other words ~0(5, L) -6~0(1, L) is additive in L on T ' for each 6 > 0. Moreover ~ (5, L) is positively homogenous in L. Thus we can state the pre- liminary result: F o r 6 > 0 a n d L E T ' : ~0 (6, L) = 6~0 (1, L) + ~ (6, L) where ~ (6, L) is a realvalued function with ;t(1, L) = 0, additive in 6 a n d linear in L.
B y use of 3.34 we see t h a t there is a function g(6) taking values from T a n d such t h a t
A (6) = 6A (1) + g(6).
Here g (6) is the element of T on which ~ (6,.) E T * * is projected. Since the projection is linear and ~ is additive in 6 it follows t h a t g is additive and g (1) = 0. N o w p u t /(5) =
= g ( 5 ) + 6a. Then f is additive so t h a t the s t a t e m e n t 1 of the theorem is proved.
The result 3 is immediate and 2 follows since a E A (1) implies /(6) = 6a § g(5) E6A (1) § g(6) = A (5).
4.4. If all the sets in a one-parameter semigroup contain the origin the semigroup is m o n o t o n e in the sense t h a t 61 < 62 implies A (61)c A (62). F o r A (62)= A (61)+
+ A (6~ - 61) and since A (62 - 51) contains the origin A (52) c A (61).
One-parameter semigroups of convex sets containing the origin are " a l m o s t "
linear. The following proposition gives t h e exact formulation.
I f B (6) is a one-parameter semigroup of convex sets containing the origin, there exist two convex sets K 1 and K 2 containing the origin and such that 6 K 1 ~ B(6) c 6 K 2. The set K 2 is obtained from K 1 by adjoining to K 1 the endpoints of all radii of K 1.
95
H. RADSTR()M,
One-parameter semigroups
Remark. B y a r a d i u s of K 1 we m e a n , of course, t h e i n t e r s e c t i o n b e t w e e n K 1 a n d a r a y from t h e origin. Such a r a d i u s is e i t h e r t h e e n t i r e r a y , t h e origin or a s e g m e n t , one e n d p o i n t of which is t h e origin a n d belongs t o K1, while t h e o t h e r e n d p o i n t m a y or m a y n o t belong t o K~.
Proo/. B y 4.1 we h a v e for r a t i o n a l r > 0 B ( r ) = r B ( 1 ) .
N o w l e t K 1 be t h e set o b t a i n e d f r o m B(1) b y d e l e t i n g f r o m B ( 1 ) all p o i n t s which a r e e n d p o i n t s of a r a d i u s a n d different from t h e origin. L e t K 2 be o b t a i n e d b y a d j o i n i n g t h e s e p o i n t s to B (1). T h u s
r K 1 c B(r) ~ r K 2.
Since B (8) is m o n o t o n e i t follows t h a t for 8 a r b i t r a r y 13 r K 1 c B ( ( ~ ) c N r K2.
r<~ r>6
I t follows f r o m t h e p r o p e r t i e s of K 1 a n d K s tha~
(J r K1 = 8 K 1
a n d f l r K 2 = 8 K s
which p r o v e s t h e p r o p o s i t i o n .
4.5. B y c o m b i n a t i o n of 4.3 a n d 4.4 i t is possible to c o m p l e m e n t t h e r e s u l t of t h e discussion in 2.2.
I / A (8) is a semigroup o/ compact, convex sets, in a locally convex, Hausdorf/, linear space, it is o/ the /orm
A(8) = / ( 8 ) + S K
where K is a compact convex set and ] (6) is a one-parameter semigroup o/points.
Proo/. L e t / be t h e f u n c t i o n of p r o p o s i t i o n 4.3, p u t B (~) = A (8) - / (6) a n d a p p l y 4.4.
w 5. One-parameter semigroups of compact sets
5.1. Theorem. I n a locally convex space any one-parameter semigroup o/compact sets consists o/convex sets.
Proo/. L e t t h e s e m i g r o u p be A (8). W e fix 0 a n d consider t h e s e t s B , = 2 " A ( ~ . ) , n = O , 1 , 2
T h e n
1 ~ B , is c o m p a c t 2 ~ B,,+I c B , 3 ~ H B , = H B o
1 ~ is obvious. 2 ~ follows f r o m t h e f o r m u l a 2 M c M + M . 3 ~ follows b y i n d u c t i o n f r o m 3.2 in t h e following w a y
ARKIV FDR MATEMATIK. B d 4 n r 9
5 = 2 " H A (~;~
H B ~ = H ( 2 " A ( : ~ =
Consider now the c o m p a c t set B = [')Bn. W e observe first t h a t B is convex.
n~0
L e t n a m e l y x, y E B . T h e n x, y C B ~ for all n. I t follows t h a t if n > ~ l we have x -:- y
x + - - Y c I - ( B , ~ + B , ~ ) = B , ~ . T h u s - - 7 ) " EB. Since B is close~l this implies t h a t B
2 2 z
is convex.
N e x t we observe t h a t B,,-~ B in H a u s d o f f f ' s topology for sets. T h a t this follows from 1 ~ a n d 2 ~ is a s t a n d a r d topological theorem.
I t follows t h a t HB~--~ H B , for the o p e r a t i o n H is a c o n t i n u o u s operation in a loca!ly convex space. (We c a n also prove t h e r e s u l t directly i n t h e following way. L e t U be a convex n e i g h b o r h o o d of t h e origin. W e n e e d t h e inequalities HB~ c- H B + U for all sufficiently large n. S i m p l y choose n so large t h a t B~ ~ B + U. T h e n b y 3.2 we o b t a i n H B ~ H B + H U . B u t U = H U since U is convex.)
C o m b i n i n g this w i t h 3 ~ above we o b t a i n H B 0 = / 1 B = B since B is convex a n d closed. On t h e other h a n d B 0 ~ B = H B 0 ~ B 0 so t h a t e q u a l i t y holds everywhere a n d it follows t h a t B o = A (5) is convex. The t h e o r e m is proved.
5.2. Strneture Theorem:
A n y one-parameter semig~vup A (5) o / c o m p a c t sets in a locally convex, Hausdor[], linear space is o/the/orm
A (5) = I(5) + 5 K
where ] (5) is a one-parameter semigroup o/points and K a compact convex set.
Proo]. T h e o r e m 5.1 a n d p r o p o s i t i o n 4.5.
R E F E R E N C E S 1. BOURBAKI, Integration. Hermann. Paris (1952).
2. GLEASON, Arcs in locally compact groups, l~roc. Nat. Acad. USA 36, pp. 663-667 (1950).
3. RAI)STRSM, Convexity and norm in topological groups. Arkiv fbr matematik 2, pp. 99-137 (1952).
4. TYC~O~OFF, Ein Fixpunktsatz. -~[ath. Ann. 111, pp. 767-776 in particular, pp. 768-769 (1935).
Tryckt den 22 januari 1960
Uppsala 1960. A]mqvist & Wiksells Boktryckeri AB
