Approximation of Weak Convergence

Math. Nachr. 160 (1993) 299-312

Approximation of Weak Convergence

By R. LOWEN of Antwerpen

(Received July 18, 1991)

1. Introduction

I t is the purpose of this paper to introduce an alternative analytical structure-a so-called approach structure [8]-on the space of all probability measures on a topological space and to submit that from the probabilistic and statistical point of view it may be a more natural structure to work with than the weak topology.

Actually, the structure which we construct and which we shall denote R is closely related not only to the weak topology, but also to the L’-metric in the sense that both these structures are derivable from 52 via the canonical categorical technique of coreflection [2]. Such coreflection in a category can be thought of as a kind of projection, however the weak topology and the L’-metric represent only partial information of R, and R itself is more than the “sum” of these two structures.

In [5] and [8] approach structures were introduced as a common generalization of topological spaces and metric spaces, based on the simple observation that in a topological space one can merely say whether a point x is contained in the closure of a set A or not, but that in a metric space one can give a numerical distance x is away from A .

In approach spaces, we axiomatize a distance between points and sets and extrapolate these ideas to other topological concepts, e.g. a distance a point is away from being a limit point of a sequence, a distance a space is from being compact (related to KURATOWSKI’S measure of non-compactness [4]), etc. ... For details we refer to [7], [8], [9].

The main purpose of this paper is first to show that R (the structure which we shall define in part 3) is a canonical structure on spaces of probability measures and second to apply the abstract theory to 0 thus demonstrating that it permits in a canonical way to measure deviation from topological behaviour in the weak topology. We look at approximate convergence of sequences of probability measures and see how this is related to the important PORTMANTEAU Theorem [l], [lo], we look at approximate convergence of degenerate measures [lo] and we study the measure of compactness of different types of contaminated sets of probability measures [3].

300 Math. Nachr. 160 (1993)

2. Preliminaries

We first recall abstract concepts required in the sequel, [5 ] , [7], [8], [9]. Let X be a set, P:= [0, co] and IP* :=lo, co[. A map 6 : X x 2' + IP is called a distance if it fulfils:

(D 1) (D 2 ) (D 3) V A , BE^^, V X E X : ~ ( X , AUB)=6(x, A)A6(x, B) (D 4) V A E 2 X , Vx EX, V E E IP : 6(x, A ) 2 6(x, A(')) + E where

vx EX : 6(x, {x}) = 0 V X E X : 6(x, 0) = co

A'":= {x E X16(x, A ) 5 E } .

A collection (@(x))~€~ of ideals in PX is called an approach system if it fulfils:

(A 1) vx E X , v v E @(x) : v(x) = 0 (A 2) VXEX, V V E I P ~ if VE, NEIP*

3 v r ~ @ ( x ) s.t. V ~ + E ~ V A N then VE@(X) (A 3) Vx E X, Vv E @(x), VN E IP* 3(~ , ) , ,~ E I7 @(.I!)

Vx, y E X : v(y) A N 5 v,(z) + v,(y) . 2eX

Each ideal @(x) is called an approach ideql (at x) and the functions in @(x) are called localdistances (at x). For simplicity of notation we usually abbreviate (@(x))~€~ to @. Ideal is meant here in the lattice-theoretical sense, i.e. a dual filter.

Distances and approach systems are equivalent concepts IS]. Thus given a distance 6 the associated approach system Gd is given by

@ , ( x ) : = { ~ € I P ~ I V ~ ~ 2 ~ : i n f v ( a ) ~ 6 ( x , A ) } XEX acA

and given the approach system the associated distance 6, is given by

6,(x, A):= sup inf v(a) x E X , A E 2". veO(x) aoA

Moreover, we always have = @ and bQd = 6, so if no confusion can occur we denote

In [8] still another way of describing approach structures was given, using a so-called regularization which unifies the operations of taking U.S.C. regularizations and non-expansive functional hulls of real valued functions (see e.g. also 1. SINGER [11], [12]). We shall not need the full machinery of this here but only the definition. I f f : X --$ IP is a map then its (@-)regularization is given by

simply @ and 6, simply 6.

J'(x):= sup i n f u + v)(y) veQ(x) yeX

and f is called (@-)regular i f p =.f.

and is usually denoted ( X , @) or simply X if no confusion can occur. A set X equipped with a distance or an approach-structure is called an approach space

If X and X ' are approach spaces andf : X + X ' thenf is called a contraction if the

Lowen, Approximation of Weak Convergence 30 1

following equivalent Conditions are fulfilled:

(C 1)

(C 2)

vx E X V V ' E @'(J(x)) : \"Of€ @(x) vx E X V A E 2 x : G'(J(x) , f (A)) _I 6(x, A ) .

Approach spaces and contractions form a topological category which we denote AP. It will be important for the sequel to know that and how the categories TOP of topologica] spaces and continuous maps and p-MET" of extended pseudometric spaces and non-expansive maps are embedded in AP. TOP is embedded bireflectively and coreflectively [2] via the embeddingsfunctor

TOP At A P

A , ( 9 ) (x):= { 11 E P"(\J(x) = 0, v U.S.C. in x} .

The associated distance 6, is given by 6,(x, A):= 0 if x E A and 6,(x, A ) = 03 if x+A. Given (X, @)EJAPI its TOP-coreflection is determined by the space ( X , T*(@)) where P(@) is the topology determined by the neighborhood system

N * ( @ ) ( x ) : = { { v < E } l v E @ ( x ) , E E P * } X E X .

If 6 is the distance associated with @, then this topology can also be characterized by its closure operator which then is given by A d = { X E X16(x, A ) = 0). Analogously p-MET" is embedded coreflectively via the embeddingsfunctor

p-MET" A, AP

( X , 4 - (A', A r n ( 4 )

where for all x E X :

A,(d) (x) := { V E I P " l V d(x;)).

The associated distance in this case is the usual distance in metric spaces, i.e. 6,(x, A ) := inf d(x, a). Given ( X , @) E IAPJ its p-MET"-coreflection is determined

by the space (A', M ( @ ) ) , where M(@) is the extended p-wetric given by

In [7] we showed that KURATOWSKI'S measure of non-compactness [4] could be recaptured as a categorical concept in AP.

In order to explain this we first however need to say what approximate convergence means in AP. This was elaborately investigated in [5 ] , [6], [9] and we now recall those concepts which are required in the sequel. It turned out that the nicest way to formulate convergence was via filters and not nets. This however need not inconvenience us, as we shall see in the sequel, whenever we start with a sequence we simply look at the FRBCHET filter generated by it.

IlEA

M(@)(x, Y):=6,(x,{Yl) v ~ , ( Y > { X ) ) .

Math. Nachr. 160 (1993) 302

If (X, @)E lAPl and 9 is a filter on X, then we define the maps a 9 and 19 in BX by

a 9 ( x ) : = sup sup inf v(y) = sup 6(x, F) YE@(X) F E F ~ E F F € F

and

19(x):= sup inf sup v(y). veO(x) FeF yeF

For each x EX, a 9 ( x ) and W ( x ) measure the “distance” fromx to an adherence point or a limit point of 9 in the TOP-coreflection of (X, @). In order to see this more clearly we refer the reader to [5 ] and [9].

A measure of the compactness of (X, @) [7] is now given by

m@(X):= sup inf W ( x ) = sup inf a 9 ( x ) * € U ( X ) xeX QeF(X) XEX

where q X ) (V(X)) is the set of all (ultra) filters on X. If a space is compact, then all ultrafilters must converge; mo(X) measures the extent to which this is the case. Not ie that if (X, @) is topological i.e. if Q, = A , ( Y ) for some topology F on X, then rn,(X) = 0 if and only if (X, 9) is compact and, analogously, if (X, @) is metric, i.e. 4, = A,(d) for some co-p-metric d on X then m,(X) = 0 if and only if (X, d) is totally bounded [7].

This measure of compactness has several other expressions and is closely related to KURATOWSKI’S original measure of non-compactness (see [4]).

3. The weak topology and the L’-metric revisited

Throughout the sequel we suppose X to be a separable metrizable topological space. F will stand for the collection of open sets, 93 for the Bore1 sets, A? for the set of probability measures on X and W for the weak topology on A. It is well known that a basis for the weak topology on A is given by the basic neighborhoods of P E A , [l], (101,

V(P, Y, E ) : = { Q E A I V G E Y: Q(G) > P(G) - E }

where Y runs through the set of finite subsets of F and &ED’*. One way to define a canonical approach structure on A? and which will, among other properties, yield the weak topology as TOP-coreflection, is the following. Rather than putting all Q E A fulfilling Q(G) > P(G) - E for all G E Y in “one bag” i.e. taking V(P, ’3, E) , we retain the numerical information P ( G ) - Q(G) truncated at 0 and taken over Y i.e. we consider the maps

d; : A + IP : Q + sup(P(G) - Q(G)) V 0 C€9s

and note that each map d” in its own right is a pseudo-quasi-metric(bounded by 1) on A. Each dp” is then a local distance giving information how far Q is away from P with respect to the collection of open sets S and it is easily verified that the family y(P):= (dgl’9 c Z Y finite} is a basis for an ideal Q(P) which makes (Q(P)),e, an approach system on 1.

Lowen, Approximation of Weak Convergence 303

When we say Y(P) is a basis for SZ(P), what we precisely mean is that

Q(P):= { V E P I V E , N EIP* 386 Y ( P ) : e + 2 A NJ

(see also condition (A 2) in the definition of approach structures). However as was shown in [7] and [8] i t suffices to work with the bases (Y(P))pcM, which of course makes calculations easier and more transparent. The distance 6, associated with the approach structure (SZ(P)),,, is then given by

b,(P, d ) = sup inf sup(P(G)-Q(G)) V 0. S ' C F QEJB GES'

9 finite

We now proceed to prove the canonicity of this approach structure on A. A first important property when dealing with the weak topology is that if f : X --f Y is a continuous map ( Y too being a separable metrizable topological space), then its natural extension

A ( X ) >A(Y)

(where & ( X ) and A( Y) are the sets of probability measures on X and Y respectively and flP) := Pof-') is continuous with respect to the weak topologies.

Proposition 3.1. If f: X --f Y is continuous then, f A ( X ) and A(Y) are equipped with their canonical approach structures, f : A ( X ) + A( Y) is a contraction.

Proof . Let 27 be a finite collection of open sets in y and P, Q E A ( X ) then

d7{p)of(Q) = SUP@(P) (GI - f ( Q ) ( G ) ) V 0 GEI

= su~(P( . f - ' (G)) - Q ( f - ' ( G ) ) ) V 0 G E I

= d?(Q)

where A?:= ( f - ' ( G ) I G ~ 2 7 e ) , and by (C 1) we are done.

Second let us now see what is the relation of D to classical structures on M.

Theorem 3.2. The TOP-coreflection of(A', SZ) is given by (A', W ) where W standsfor the weak topology on .X.

Proof. It is immediate to verify that if PE&, Y c 5 is finite and c > 0 then { d : < E } = V(P, 27, E ) .

The result thus follows from thc characterization of the TOP-coreflection of an approach space given in the preliminaries.

Theorem 3.3. The p-MET"-coreflection of(&, 52) is given by (A!, d ) where

d(P, Q):= SUP IP(G) - Q(G)/ G E S

is the L'-metric on A.

Math. Nachr. 160 (1993) 304

Proof . This follows at once from the definition of the distance 6, associated with 51 and from the characterization of the p-MET"-coreflection of an approach system given in the preliminaries.

The foregoing two theorems enhance the naturality of l2 since two important structures on M are obtained from 51 via purely categorical considerations. From 3.1, 3.2, and 3.3 we also immediately derive the following result.

Corollary 3.4 .1fJ: X -+ Y is continuous thenp is continuous w.r.t. the weak topologies

I t is well-known (see e.g. F. Tops0E [13]) that the weak topology is the coarsest and is non-expansive w.r.t. the L'-metrics.

topology making all maps

w c : M + P:P 3 P(G), G E F ,

upper semi-continuous, i.e. such that they coincide with their U.S.C. regularizations.

introduce it is given by the following result. A third argument to stress the canonicity of 51 and which gives another way to

Theorem 3.5. 51 is the coarsest approach structure on M making all maps oG, G E 9, regular.

P r o o f . By definition the regularization (of wG, taken in P E A , is given by

Qg(P) = sup inf (wc + dF)(Q) 9,Cg Q d i

%finite

- 2 inf ( Q ( G ) + (P(G) - Q(G)) V 0) QEJC

2 P(G) = o J P ) .

Since we always have 6 g 5 wc this shows that indeed toc is regular. Now if A is an approach structure on &'such that all wG, GEY, are regular then for P ~ d d we have

o c ( P ) 2 sup inf (wG + I) (Q) I c A ( P ) QEM

i.e. Vc > 0 3A E A ( P ) VQ E A! :

wc(P) 2 (oc(Q) + 4Q) + which implies that dp" done. R

A + E. Consequently by (A 2) we have dFEA(P) and we are

4. Convergence in (A, Q)

Throughout the sequel, if (Pn)nEN is a sequence of probability measures then (P,) will stand for the FRBCHET filter generated by it. We shall put I, for the "convergence measure" in (M, SZ) as defined in the preliminaries (see also [Sj, [6], [9]).


Proposition 4.1. lf (PJn is a sequence of probability measures in At and P E .At then & ( ( P n ) ) ( P ) I a f and only f for each Y c 5 finite and each p > a we have V(P, $9 P I E Vn>.

P r o o f . This follows at once from the observation that

I Q ( ( P n ) ) ( P ) = sup inf sup sup(P(G) - Pk(G)) V 0 , 9 C F neN k z n G E ~

%?finite

and a straightforward verification.

BILLINGSLEY [l] or PARTHASARATHY {lo], the following result. Making use of this proposition one then obtains, along the same lines as in

Theorem 4.2. (PORTMANTEAU theorem). I f (P,,),, is a sequence ofprobability measures and P E A then

AQ((Pn)) (P) = s u p K ( P ( G ) - P,(G)) V 0 9€F n

= sup lim(P,(F) - P(F)) v 0 Fclosed n

- = sup lim IP(A) - P,(A)I

= sup lim l j fdP - jSdP,I

= sup lim IjW - jfdPnI.

AP-continuity s~ n -

f continuous n o j / j 1

- f uniformly n continuous

I

P r o o f . For simplicity in notation we put l:=A,((Pn))(P) and pl , ... , p 5 the five expressions for I given above. That i = p1 = p2 follows at once from Proposition 4.1 making V(P, G , /I) E (P , ) explicit and applying complementation. That p 3 5 I follows from /z = p 1 = p 2 and from noting that if A is a P-continuity set then

l im(P, (A) - P ( A ) ) v 0 s lim(P,(A) - P(A)) v 0 n n

and - lim ( P ( A ) - P,(A)) v o 5 l i m ( ~ ( A ) - P,(A)) v 0 .

n n

Conversely, if F is closed, then take a sequence S, lO such that for all k

F k : = { X E X l d ( X , F ) s 6 k }

( d any metric that metrizes .F) is a P-continuity set. Then - lim (Pn(F) - P(F)) v 0 5 sup lim (Pn(Fk) - P(Fk)I

n k n

which shows 1 s p 3 .

20 Math. Nachr., Ed. 160

306 Math. Nachr. 160 (1993)

Next iff is continuous, take k E N o and put

then - 1 l k - lim (J jdP , - JjilP) V 0 5 - + - 1 lim (P,,(F:) - P(F:))

n k k , , l

which by the arbitrariness of k and upon applying the same reasoning to 1 -fproves p4 5 I,. That p5 _I p4 is clear.

Finally if F is closed then by regularity, for any E > 0 there exists 6 > 0 such that if we Put

G6:= { X E X ( d ( x , F ) < 6 ) , then P(G,) < P(F) + E . Now if we put

f ( x ) : = ( l - , d ( x , F ) ) 1 V 0 ,

then clearly f is uniformly continuous on X and - lim (P,,(F) - P(F)) V 0 5 K ( J f d P n - P(F)) V 0

n n

5 (P5 + JfdP - P(F)) v 0

I Ps + P(G6) - P(F) s P5 + E

which by the arbitrariness of F and E proves I S p 5 and we are done. This result shows that convergence in s2 measures a deviation of convergence in the

weak topology in the sense that the smaller the value of I Q ( ( P , , ) ) ( P ) the more (P,,!,, approximates being weakly convergent to P and where “true” weak convergence IS obtained if &((P,,))(P) = 0.

To further illustrate the way this works we look at degenerate measures. It is well-known that X can be embedded as a closed subset of A’ with regard to the weak topology (see e.g. PARTHASARATHY [ 101) via the map X + A : x -, P, where P, stands for the degenerate probability measure at x . That X is embedded as a closed subset of A? means that sequences of degenerate measures, if they weakly converge, can only do so to a degenerate measure.

The following result shows how much more discernitive s2 is when compared to W.

Theorem 4.3.1fV stands for the collection of degenerate probability measures on X and P E A then

6,(p, Y ) = 1 - sup P ( { x } ) . *EX

P r o o f . We shall only treat the case where P has a countably infinite set of atoms. The other cases are easy modifications and are left to the reader.


Let a be the P-measure of an atom a e X of largest measure. Then

4dp, yr) 6 &v, { P a } ) = SUP SUP(P(G) - P,(G)) V 0

= SUP (P(G) - P,(G)) V 0 O C F G&

O f ini le

O E 9

= P(X\a) - P,(X\a) = 1 - a .

Conversely, if 9 c F is finite, then

inf sup(P(G) - P,(G)) V 0 = 0 xaX CEO

if 09 # 0 and thus

&(P, V ) = sup inf sup P(G) O C S f i n i l e x e X GEO

n O = 0 x+G

= sup inf inf supP(G)

2 - sup inf inf P(H)

= sup inf P(G).

VCFf in i t e HE% xeX\H Gag n%=0 XW

O C J f i n i t e HE% xeX\H n ~ = 0

S C T f i n i t c CEO n O = 0

Now let us suppose the set of atoms { a , ( n ~ N } ordered in the sense that P({a,+ 2 P((an}) for all EN. Then a = P ( { a o } ) and #l:=max({P({x}) lxeX}\{a}) , i.e. P is the measure of a second largest atom. Let ~ ~ 1 0 , a - #l[ and take n0€N such that

& C p({ an}) < 2 . n > n o

Next choose r > 0 such that for all n E (0, ..., no}

P(B*(an, r ) ) 2 P({an}) + C .

Put l’:=X\{a,(ii~IN} and partition I’in disjoint B O R E L S ~ ~ S D, , ..., D,, such that for all i ~ { l , ..., nl}

P(Di) < P Next choose 6 > 0 such that

& n , S S - .

2 Since P is regular we can find closed sets K i c Di such that for all irz (0, ..., n,}

P(Ki) s P(Di) 5 P ( K i ) + 6. Since the sets K O , . . ., K,,, {an(n 5 no} are pairwise disjoint and closed it follows from the normality of X and the regularity of P that we can find open sets Oi 3 K i such that the following properties are fulfilled:

20‘

308 Math. Nachr. 160 (1993)

(i) 0,, . . . , Onl are pairwise disjoint (ii) for all ie{O, ..., n l } : P ( K i ) ~ P ( O i ) ~ P ( K i ) + 6

(iii) for all i e (0, ..., n , } : oin{anln =( no} = 0 Now consider the following finite collection of open sets

no ( 5 ) " I Gi:= u OjU u B ak, - i E { O , ..., n,}

j = O k = O J # i

Hi:=X\B*(ai, r ) i E ( 0 , ..., no}

Then by construction

For each ~ E { O , ..., n,} we further have

= 1 - E - P(Di) 21 --E-DZI - - c1

and for each i E (0,. . . , no} we have

P(Hi ) = I - P(B*(ai, v ) ) 21 - - c I - & .

Consequently if we put

9e,:= {GiliE (0 ,..., n , } } U{Hi l i~ (0 ,..., no}} then

inf P(G) 2 1 - a - E G€SU,


From the arbitrariness of E > 0 it then follows that

b,(P, V ) 2 sup inf P(G) 2 1 - OL BCFfini te G e l

and we are done. For the weak topology the “distance” of P to Y r is 0 if P E V and co if P 4 Y. So

even if P = (1 - E ) P, + EQ where E can be arbitrarily small, P is never “topologically close” to V . 52 on the other hand recognizes that P is statistically similar to an element in V and takes this similarity into consideration when giving the distance pis away from being in the closure of V. From the general theory of convergence in AP (see [ 5 ] , [6] , [9]) we know that distance can also be derived from convergence by a sjmple formula which in our case yields

b,(P, V ) = inf in .F(P) , S a f i l ler on& containing V

The interpretation is that for no sequence (filter) in K9’ comes closer to being a limit of that sequence (filter) than

b,(P, V ) = 1 - ( I - & ) = & ,

and of course there is at least one sequence for which the distance is precisely E : the constant sequence (P , = PJ,!

5. Compactness in (A, 52)

In statistical applications (see e.g. P. HUBER [3]) one is often confronted with “contaminated sets” of probability measures. If .# is a “nice” set of probability measures then it may become “contaminated by a set X c A to an extent E > 0” in the sense that one has to consider the new set

%:= ((1 - E ) P + E Q ( P E Yt, Q E if}

or one may have to consider

L Y : = ( Q E M ~ ~ P E X ‘ : IIP-QII SC}

where 1 1 I( stands for the L’-metric given by I1P - QII :=sup IP(B) - Q(B)I. In both 8646

cases it is clear that even if 2 is weakly compact, neither 8; nor 9 need be weakly compact or even weakly relatively compact in general. However for E sufficiently small, i.e. if the masses which may “escape to infinity” are negligible, 3 and 9 may “statistically” still behave as weakly compact sets. We shall now see that in SZ the measure of compactness [7] gives us precisely this information.

Math. Nachr. 160 (1993) 310

Theorem 5.1. I f 3 c A is weakly compact, X c A is arbitrary and E > 0 isfixed and we put

y:={(l - E ) P + E Q ~ P E ~ , Q E X }

then m,(J) S 2 ~ .

P r o o f . The compactnessmeasure of 2 (see preliminaries or 171) is given by

Now if F is a filter on 2 and 9 E F then put c

%:={PE&')3QEX:(l - & ~ ) P + E Q E ~ }

Then it is immediately verified that

P:= (($19 E F } )

is a filter on X'.

Claim: For any filter F on 9, P E .% and Q E X :

cr,F((l - c) P + E Q ) 5 a, P(P) + 2~

Indeed let us put R:= (1 - E ) P + EQ then

a z ( R ) = sup sup inf sup ( R ( G ) - qG)) V 0 9 C S F e F T e 9 G E I d f inite

Fix a finite collection of open sets Y c 9, a % E F and a TE 9. Let U E Yf and VEX be such that T = (1 - E ) U + EV then

SUP (R(G) - T(G)) V 0 = SUP (( 1 - E ) (P( G ) - U ( G)) + E(Q( G) - V'(G))) V 0 G E 9 G € I

5 - SUP (P(G) - U(G)) V 0 + 28 G€d

and consequently

a,F(R) S sup sup inf sup (P(G) - U ( G ) ) V 0 + 2-5 I C Y FeF U E F Ged 9 f inite

= a&P) + 2 E .

Now since X' is weakly compact and since by Theorem 3.2 the weak topology is the TOP-coreflection of 0 it follows from Theorem 4.3 in [7] that m,(X') = 0. It then follows that

Lowen, Approximation of Weak Convergence 31 1

and we are done.

Theorem 5.2. If X c A? is weakly compact, E > 0 is fixed and we put

$ : = { Q E A ? ~ ~ P E X : IIP-QI( sc} then ma($) 5 2c.

Proof . The proof goes along the same lines as the one of 5.1. Let U be an ultrafilter on d . For each Q E f choose a fixed P, E X such that IIP, - QII 5 E and such that P , = Q if Q E X . Then for each % E U put

@ : = { P Q ~ Q E % } .

It is immediate that ~ : = ( { & I % E V } ) is an ultrafilter on Z. Now if Q E ~ then

A, u(Q) ILQ ~ ( P Q ) -k 2 6 .

Indeed this follows at once from the fact that for any Ok E U, Y c 9 finite, R E % and G E $9 we have

(Q(G) - R(G)) V 0 2 (PO(G) + E - PR(G) + E ) V 0 5 (Pp(G) - PR(G)) V 0 + 2 ~ .

Consequently we also have

m , ( f ) g m,(X) + 2E = 2 E

and we are done.

References

[I] P. BILLINGSLEY, Convergence of probability measures, John Wiley & Sons, 1968. [2] H . HERRLICH and G . STRECKER, Category theory 2nd cd., Heldermann Verlag. [3] P. J. HUBER, Theorie de I’inference statistique robuste, Publ. Sem. Math. Supkrieures 31, 1968,

Presses de I’Universite de Montreal. [4] C. KURATOWSKI, Sur les espaces complets, Fund. Math. 15 (1930), 301-309. [5] E. LOWEN and R. LOWEN, A quasitopos containing CONV and MET as full subcategories, Int. J.

Math. & Math. Sci. 11 (1988), 417-438. [6] E. LOWEN and R . LOWEN, Topological quasitopos hulls of categories containing topological and

metric objects, Cahiers Top. Gtom. Dim., Cahiers Top. Gtom. DiIT. 30 (1989), 2/3-228 [7] R. LOWEN, KURATOWSKI’S measure of non-compactness revisited, Quart. J. Math. Oxford 39 (1988),

235-254.

Math. Nachr. 160 (1993) 312

[8] R . LOWEN, Approach spaces :a common supercategory of TOP and MET, Math. Nachr., 141 (1989),

[9] R . LOWEN, A topological category suited for approximation theory? J . Approximation Theory, 56

[lo] K. A. PARTHASARATHY, Probability measures on metric spaces, Academic Press 1967. [I 11 1. SINGER, Conjugation operators, LNEMS 226 (1984), 80-97 Springer. [ 121 I . SINGER, Some relations between dualities polarities, coupling functionals and conjugations,

J . Math. Anal. Appl. 115 (1986), 1-22. [13] F. TOPSQE, Topology and measure, Lect. Notes in Math. 133, Springer, 1970.

183-226.

(1989), 108-1 17.

Departement Weskunde en Inforniatica Universitair Centrum Antwerpen (RUCA) Universiteit Antwerpen Groerienborgerlaan I 7 1 2020 Antwerpen Belgium

Documents

Approximation of Weak Convergence