Armstrong, Thomas E.

Sudderth, William

University of Minnesota Digital Conservancy

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On Nearly Strategic Measures 
by 
Thomas E. Armstrong* and William D. Sudderth* 
Technical Report No. 334 
*Research supported by National Science Foundations Grants MCS 74-05786-A02 
(for Armstrong) and MCS 77-28424 (for Sudderth). 
• 
~ 
V 
Abstract 
Every finitely additive probability measure a defined on all sub-
sets of a product space Xx Y can be written as a unique convex 
combination a= Pu+ (1-p)v where µ is uniformly approximable by 
strategic measures and v is singular with respect to every strategic 
measure. 
AMS 1970 subject classifications, 60005, 60B05, 28A60, 46A4o. 
Key words: finite additivity, strategy, conditional probability, 
decomposition of measures, convex direct sum, split face, Choquet simplex, 
disintegration. 
1. Introduction. For each nonempty set X, let P(X) be the collection 
of finitely additive probability measures defined on all subsets of 
X. A conditional probability on a set Y given X is a mapping from 
X to P(Y). A strategy a on Xx Y is a pair (cr
0
,cr1) where 00 is 
in P(X) and cr1 is a conditional probability on Y given x. Each 
strategy a on Xx Y determines a strategic measure, also denoted 
a , in P = P(X x Y) by the formula 
where g is a bounded, real-valued function on Xx Y. The collection 
L of all strategic measures was studied by Lester Dubins [5], who proved 
that, if X or Y is finite, then every member of P is nearly strategic 
in the sense that it can be uniformly approximated arbitrarily well by 
a strategic measure. Although strategic measures are the natural objects 
in gambling theory (Oubins and Savage [6]), the 
collection N of all nearly strategic measures is more tractable than 
L for some purposes. As evidence, witness the fact that N is always 
convex (Proposition 3.1), whereas L need not be (Example 3.1). Indeed, 
one would be tempted to restrict attention to N had not Dubins [5] also 
shown that, if X or Y is infinite, then there exist elements in 
",... J. ( -- N .l) 
_ _, , the set of measures in p singular with respect to every 
measure in~. (As usual the finitely additive probability measures u 
and v are singular, written u J. v, if, for every positive t , there 
is a set A such that µ(A) < e: and v(A) > 1-e.) 
For the statement of the main result, define a convex set K to 
be the convex direct sum of two disjoint, convex subsets A and B, 
written K = A (f- B, if every element x of K can be expressed as a 
-1-
convex combination 
(1.2) X = pa + ( 1-p) b 
with a e A, be B, and O ~ p ~ 1 where pa, (1-p)b, and pare unique. 
The sets A and B are convex direct summands of K. 
Theorem 1. 1. The sets ~J. N and L..J are convex and disjoint, and 
P = Ni+'2/·. 
As mentioned by Dubins [5], it follows from results of Bochner and 
Phillips ['3] that 
(1 "') 
' . .) 
Since /J .lJ. = N 1..1.:::, N, it is clear from {1.3) and Theorem 1.1 that l: J.J. ::!: N, 
which answers a question posed by Dubins. 
The proof of Theorem 1.1 will be based on a characterization of the 
convex direct summands of P. This characterization is valid when P 
is the collection P(B) of all finitely additive probability measures 
on an arbitrary Boolean algebra A and, in particular, when ~ is the 
algebra of our main interest, that of all subsets of Xx Y. Two 
definitions are needed for the characterization. 
A face of a convex set K is a convex subset F which contains the 
endpoints of a line segment [a,b] = {ta + (1-t)b: 0 ~ t ~ 1} c I{ whe_never it 
contains an interior point ta+ (1-t)b with O < t < 1. A convex sub-
set K of a linear topological space is a-convex if, given x1 ,x2 , ••• in 
K and nonnegative real numbers Q'l ,o-2 , such that 2.ot = 1, then the n 
series L} Q' x converges to a point in K. The norm topology on P = P( !R) 
n n 
is the topology from the usual norm defined by 
(1.4) 111-L - vi I = s u P f. f 1-L ( B ) - v( B) I : B e ~ 1 , 
for u, v e P • 
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Theorem 1. 2. If A is a face of P, then the following conditions are 
equivalent: 
(a) A is a convex direct summand of P. 
(b) A is norm-closed. 
(c) A is cr-convex. 
Furthermore, if A is a convex direct summand of P, then P = A r+>AL. 
'!he proof of Theorem 1.2, which is given in the next section, is 
based on a characterization of the convex direct summands of abstract 
Choquet simplexes due to Goodearl [8] and Lima [9]. In sections 3 and 
4, it will be shown that N is a norm-closed face of P which together 
with Theorem 1.2 implies Theorem 1.1 Section 4 contains a generalization 
of Theorem 1.1 which treats nearly disintegrable measures. 
This section concludes with two well-known examples of convex direct 
sum decompositions of P. 
For u,v in P, vis absolutely continuous with respect to~' 
written v << u, if, for every € > O, there is a a> 0 such that, for 
all B g 8, u(B) <.~implies v(B) < e• 
Example 1.1. (Bochner and Phillips [3]) Let LL e P and define 
A = [ 'J e P : v << U} • 
LL 
Then P = A (±)Al. where 
LL U 
AJ. = [v e P: 
u 
\)J. µ,}, 
Example 1.2. (Yosida and Hewitt (10]) Let A be the collection of 
countably additive measures in P. Then Al. is the set of purely finitely 
additive measures in P and P = A (f)AL. 
Let K be a convex subset of a locally convex, Hausdorff, linear 
topological space E. Every subset S of K is contained in a smallest 
face, face (s), of K. Indeed, face (s) is just the intersection of all 
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> I 
faces containing s. For x r, K, face ([x1) is abbreviated to face (x). 
To each subset S of K is associated the set s' complementary t~ 
s which is defined to be the union of all faces of K not intersecting 
face (s). It is now possible to define the fundamental notion of this 
section. 
Definition. Let F be a face of K. Then F is said to be a split 
face of K if F' is a face of K and if every element of K-(FUF') 
can be written in one and only one way as a convex combination of an 
element in F and an element in F '. 
The relevance of split faces in the present context is clear from 
the following lemma. 
Lemma 2.1. Suppose A and B are disjoint convex subsets of K such 
that K=A.-t1B. Then A isasplitfaceand B=A'. 
Proof: It is straightforward to check that A and B are faces and 
that A'= B. Hence A is a split face. r· 
Thus Theorem 1.2 can be viewed, in the light of Lemma 2.1, as a 
characterization of the split faces of P. Such characterizations have 
been given for the probabilities on a compact space by Lima [9] and for Choquet 
simplexes by Alfsen [l] and Goodearl [8]. It is possible to deduce our 
results from those of Lima by using a famous theorem of Stone to represent 
each finitely additive probability on a as a Radon measure on the 
Stone space of 9. It would also be possible to develop the theory 
directly. We will instead take what seems to be the shortest route to 
Theorem 1.2 using the theory already developed for Choquet simplexes. 
We will not take any unnecessary detours into the theory of split faces 
but refer the interested reader to the papers mentioned above and to 
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additional work by Alfsen and Schultz [2] and Ellis [7]. 
If K is a compact, convex set which is the base of the positive 
cone E+ of the locally convex space E = E+ E+ and if E is a lattice 
+ 
when ordered by E, then K is said to be a Choquet simplex. The 
space E becomes a Banach space when it is given the norm which has the 
convex hull of K u(-K) as its unit ball. When endowed with the weak 
topology, the space P = P(a) is a Choquet simplex in the vector space 
BA(~) of finitely additive, signed measures of bounded variation defined 
on s. In fact, P is the base of the positive cone BA+(q) of positive 
measures in BA(~). 'fue norm on BA(3) associated with P is the 
variation norm for finitely additive measures. 
Theorem 2.1. (Goodearl (8, Theorem 9]). If F is a face of a Choquet 
simplex, then the following are equivalent: 
(a) F is a split face. 
(b) F is norm closed. 
(c) F is er-convex. 
Except for its final statement, Theorem 1.2 now follows from Lemma 2.1 
and Theorem 2.1. The proof of Theorem 1.2 will be complete once it is 
verified that, for any split face F of P, F' = Fi. The proof of 
this equality will be given in the lemmas below which are based on the 
work of Bochner and Phillips [3] and that of Goodearl [8]. 
For U, e P, define A = f v: v << ul• 
u 
Lemma 2.2. A is a split face and A' = A1 
u u u 
Proof: Use Example 1.1 and Lemma 2.1. Q 
Lemma 2.3. If u e P(~), then face (u) is the set of all v e P(s) such 
that \J s:; Au for some ~ ?- 0. 
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Proof: The collection of such vis easily seen to be a face containing 
face (u)• On the other hand, suppose v ~ AU for some A~ O. Assume 
without loss of generality that }, > 1. Let 
= \u, - V 
O! A - 1 • 
Then 
u = \ -l v + (1 - A -l)O! 
from which it follows that v e face ~u)• [1 
If ~ e P(a), then a u-density is a bounded, nonnegative function 
f whose u-integral is well-defined and equal to one. ( In the case 
considered in subsequent sections, a is the set of all subsets of 
a set X and every bounded function on X is u-integrable.) To each 
u-integrable function f is associated a measure fdµ in BA(~) whose 
value at Bee is ffdu. 
B 
Corollary 2.3. If u ~ P~a) and f is a u-density, then fdµ e face (u). 
A subset B of P is closed with respect to absolute continuity 
if A c B whenever u e B. 
u. 
Lemma 2.4. Let B be a subset of P which is norm closed and contains 
fd 11 whenever µ e B and f is a nonnegative, a-simple function with 
u-integral one. Then B is closed with respect to absolute continuity. 
Proof: Let u ,. B, and \> e A • µ, By the finitely additive Radon-Nikodym 
Theorem (Dubins [4]), there is, for each e > O, a a-simple function f 
such that l!·v - fdµ,jj < €• It is easy to see that f can be taken to be 
nonnegative with u-integral one. By hypothesis, £due B. Since B is 
norm closed, v e B. D 
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-. 
Lemma 2.5. Split faces of P are closed with respect to absolute 
continuity. 
Proof: Apply corollary 2.3 and Lermna 2.4. O 
Lennna 2.6. If F is a split face of P, then F' = Fi. 
Proof: It will first be shown that F.L c F'. To this end, let 
Since P = F Et) F ', µ, can be written in the form 
u = Pv + (1-p)v' 
.L µ, e F • 
where v e F, v' e F ', and O ~ p ~ 1. Hence, the measure p v is both 
singular and absolutely continuous with respect to u• Consequently, 
p = 0 and u e F '. 
For the opposite inclusion, let u c F' and v e F. It suffices 
to show u .L v • By Example 1. 1 
u = Pv + (1-p)v 
a s 
where By Lennna 2. 5, "a I F • 
If p ip O, then "a e A· and, by Lemma 2.5 again, "a e F"._ Hence, 
U, 
p must equal zero, and u .L v. 0 
The proof of Theorem 1.2 is now complete. 
It is convenient, in concluding this section, to present one 
additional lemma. 
Lemma 2.7. If F is a convex subset of P and is closed with respect 
to absolute continuity, then F is a face. 
Proof: Let u = pv + (1-p)v' e F and suppose O < p < 1. Then 
( v, v '} c A c F. D 
u 
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It will be shown in the next two sections that the set N of 
nearly strategic measures is a split face of P. 
3. The set of nearly strategic measures is convex. 
In the remainder of the paper, P = P(X x Y), ~ , and N are 
as defined in the introduction. The object in this section is to 
prove the following result. 
Proposition 3.1. The set N is convex. 
The proof is based on three lemmas, the first of which is due to 
oubins. To state it, associate to each as P(X x z) its marginal 
a
0 
e P(X) which is defined, as usual, by Ql
0
(E) = Ql(E x z) for E c x. 
Lemma 3.1. Suppose Z is a finite set, QI s P(X x z), and i > O. 
Then there is a strategy ~ on Xx Z such that ~o = a
0 
and 
Ila - ~II < e-
Proof: This lemma is a special case of Dubins [5, Proposition l]. CJ 
Lemma 3.2. Let a, ,. s ~ and O ~ p s 1. Then the measure 
~=Pa+ (1-p)T is in N. 
Proof: Let e > O. It suffices to find \J e :E such that 
(3.1) II µ, - vi I ~ e • 
Define v
0 
= µ,
0
; that is, v = Po + (l-p)1. 
0 0 O 
To define 
first let z = (0,1} and consider the strategy ~ on z x X which 
has a 'I' • 
0 
Next consider 
the measure QI on Xx Z obtained from X by reversing the coordinates; 
in other terms, for each bounded, real-valued function g on Xx z, 
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0 
O'g = fg where g(z,x) = g(x,z). Notice that 
Q' = Pcr + ( 1-p) 'T' = " • 0 0 0 0 
Apply Lemma 3.1 to obtain a strategy ~ on Xx Z with 
u. 2) so = Q'o, IIQ' - i,11 < e. 
Now define 
"l(x) = ~1(x)((O})cr1(x) + ~1(x)((l})T1(x) 
for each x e X. It remains to verify (3.l). 
To that end, let Ac Xx Y and define g:X x Z-+ [0,1] by 
g(x,O) = a1(x)(Ax), g(x,l) = T1(x)(Ax). 
It follows from (3.2) that 
(3.3) IQ'g - ,gl ~ ~-
However, 
(3.4) ag =Ag= JJg(x,z)dA 1(xlz)dA 0 (z) 
and 
= p J cr 1 ( X )(Ax) dcr O ( X) + ( 1-p ) f T 1 ( X) ( Ax) d ,. 0 ( X) 
= Pcr(A) + (1-p)~(A) 
= µ,(A)' 
(3.5) ~g = fJg(x,z)d~ 1(zlx)d~ 0 (x) 
= J[~ 1(x)([O})g(x,o) + ~1(x)(fl})g(x,l)]d~ 0 (x) 
= J"i (x)(Ax)d"0 (x) 
= "(A). 
Because A is an arbitrary subset of Xx Y, the desired inequality 
(3.1) now follows from (3.3), (3.4), and (3.5). C 
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The next lemma is just a restatement of the definition of N. 
Lemma 3. 3. N is the closure of ~ in the norm topology. 
Proof of Proposition 3.l: It follows from Lenuna 3.2 that N contains 
the convex hull of ~ • It then follows from Lemma 3.3 that N is 
the closure of the convex hull of ~ and, hence, is a convex set. D 
In contrast to Proposition 3.1, the set 6 need not be convex as 
this example demonstrates. As a result, L need not be norm closed. 
Example 3.1. Let X = {1,2,···}; let Y = (0, 1}; let cr e E be such 
that cro(fx}) = 0 and 0-1 (x) = a(o) for all X e X; let Te E be 
such that T ( (x}) >0 and Tl ( X) = 8( 1 ) for all X s X; define 
0 
p = \cr + \-r. Then p +6· To see this, suppose to the contrary 
p is strategic. Then, for each x e X, 
T0 ((x}) = ~({(x,l)})= 2p({(x,l)}) = 2p 1(x)((l})p 0 ((x}) 
= p1(x)((l})T0 (fx}) 
so that p1(x) = 5(1) for all x. Hence, 
= 0, 
a contradiction. 
4. The set of nearly strategic measures is closed with respect to 
absolute continuity. 
As indicated by the title, the following proposition is proved 
in this section. 
Proposition 4.1. If u e N and ·" << u, then " e N. 
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that 
0 
This proposition, together with Proposition 3.1, Lemmas 2.7 and 
3.3, and Theorem 1.2, implie·8i Theorem 1.1, the main result of the 
paper. The proof of the proposition may be of independent interest. 
For it is based on the following lemma, which may be viewed as a 
version of Bayes formula for strategic measures. 
Lemm.a 4.1. If cr e ~ and f is a a-density, then \J = fdcr e Z: • 
Indeed, if g(x) = J£(x,y)dcr1 (y Ix), then \J is the strategy ( \>0 , \J1 ) 
( \ -\)l XJ - f(x, • ) c I ) g(x) dcr1 • x if g(x) > O, 
and '\J1(x) is an arbitrary probability measure on Y if g(x) = 0. 
Proof: Let B = [x e X: g(x) > O}. It is easy to verify that 
'\J
0
(B) = 1. Now let ~ be a bounded function on Xx Y and calculate 
as follows: 
Lemma 4.2. If u e N and f is a µ-density, then fdµ e N. 
Proof: Let e > 0. Because µ, e N, there is a cr c ~ such that 
IIµ, - crll ~ e sup£. 
Hence, 
llfdµ, - fdcrll ~ e 
and, in particular, 
-11-
from which it easily folloNS that, for g = £/ rfdcr, 
u 
Thus !lfd.1 ·• gd:r\l :s: 2~, and, bP.cause g is a 0 -density, gdcr € L 
by Lemma 4.1. Hence, fdu ~ N by Lemma 3.3. C 
Proposition 4.1 now follows from Lemmas 4.2, 3.3, and 2.4. The 
proof of Theorem 1.1 is also complete now. 
The assufflption made in this paper that densities are bounded can be 
dispensed with in Lemmas 4.1 and 4.2 if. the integral of a nonnegative 
function is defined as the supremum of the integrals of the bounded 
functions which it majorizes. 
Nearly disintegrable measures. 
Let CT be a collection fsx: x e X} of nonempty subsets of Y. 
A measure u R P(Y) is r,-disintegrable if there is a pair (cr
0
,cr1) 
such that CTO ~ P(X), a1 (x) € P( Sx) for all x, and 
u(A) = Ja1(x)(Ansx)dcr0 (x) 
for every A C: y. The set N(G) of nearly G-disintegrable measures 
is the norm closure in P(Y) of the set D(G) of u-disintegrable 
measures. Here is g result which extends Theorem 1.1 to this new setting. 
Theorem 5.1. P(Y) = N(G) + D(u)~ 
It is trivial that N(G)~ = D(u)~ so that Theorem 5.1 is equivalent 
to the assertion that N(G) is a split face of P(Y). The rest of 
this section is devoted to proving the latter fact. The main idea of 
the proof is to associate with D(G) a certain collection of strategic 
measures on Xx Y. 
Lemma 5.1. Suppose A c X x Y, cr e ~, and cr(A) > 0. Then the measure 
-1 
v = a(A) lAdcr corresponds to a strategy (v
0
,v1) satisfying 
v1(x) (Ax)= k for all x. 
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~· 
• 
Proof: Apply Lemma 4.1.0 
Let ~ be that mapping from P = P(X x Y) onto P(Y) which sends 
each measure to its marginal on the second coordinate; that is, 
~(µ)(A)= µ(Xx A) 
for u e P, Ac Y. Let S be that subset of Xx Y given by 
S = {(x,y); ye sx}, and define PS=[~ e P: µ(S) = l} 
and ~s = E n P s. 
Lemma 5.2. ~(~) = D(Q). 
Proof: That ~~S) contains D(G) is an easy consequence of the 
definitions. The reverse inclusion uses Lemma 5.1. D 
Let N8 = N n P8• Use A to denote the norm closure of a set 
A of measures. 
Proof: The closed set NS contains ~S and, therefore, contains 
its closure. For the opposite inclusion, let ~~NS. Then there 
exists a sequence of strategies a e E which converge in norm to 
n 
Since u(S) = 1, crn(S) converges to 1. It follows that the measures 
-1 
·v = cr ( S) 18dcr also converge in norm to u• But by Lemma 5.1, n n n 
vn e ~S for all n. O 
Lemma 5.4. N(G) is convex. 
Proof: calculate as follows: 
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The successive steps are, respectively, by definition of N(G), Lemma 5.2, 
the norm continuity of ~, because NS ::JL'g, and by Lennna 5.2 again. 
Take norm closures above to get 
N(G) = cp(N8). 
Moreover, ~ is an affine mapping and NS is convex since it is the 
intersection of the convex sets N and Ps· Therefore, ~(NS) is 
convex as is its closure N(a). O 
Lemma 5.5. If µ e D(G) and f is a µ-density, then fdµ e D(a). 
Proof: By Lennna 5.2, there is a cr e ~ such that ~(cr) = µ• Let 
g(x,y) = f(y) for all x,y. Then g is a a-density. By Lemma 4.1, 
gdcr e P8• Thus gdcr e ~S and fdµ = ~(gdcr) is in o(a) by Lemma 5.2. D 
Lemma 5.6. If µ e N(G) and f is a µ-density, then fdµ e N(a). 
Proof: Easy using Lemma 5.5 and similar to Lemma 4.2. D 
It now follows from Lemmas 2.4 and 5.6 that N(G) is closed with 
respect to absolute continuity. Lemmas 2.7 and 5.4 then imply that 
N(a) is a face. By definition, N(G) is norm closed. Theorem 5.1 
now follows from Theorem 1.2. 
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.. 
t 
j 
~ 
References 
1. Alfsen, Erik M. (1965), "On the decomposition of a Choquet simplex 
into a direct sum of complementary faces", Math. Scand., 17, 169-176. 
2. Alfsen, E.M., and Schultz, F.w. "On Non-Commutative Spectral theory 
and Jordan algebras", (to appear). 
3. Bo:::hner, s. and Phillips, R. s. ( 1941), "Additive set functions 
and vector lattices", Annals of Mathematics, 42, 316-324. 
4. Oubins, L.E. (1969), "An elementary proof of Bochner's finitely 
additive Radon-Nik.-,dym theorem", American Mathematical Monthly, 
76, 520-523. 
5. Oubins, Lester E. ( 1975), "Finitely additive conditional probabilities, 
conglomerability, and disintegrations", Annals of Probability. 
3, 89-99. 
6. Dubins, Lester E. and Savage, Leonard J. (1976). HO;! to Gamble 
If You Must: Inequalities for Stochastic Processes, Dover, New York 
7. Ellis, A. J. ( 1977), "A facia 1 characterization of Choquet simplexes 11 , 
Bull. London Math. Socie_;x_, 9, 326-327. 
8. Goodearl, K.R. ( 1976), "Choquet simplexes and a-co1.wex faces", 
Pacific Journal of Mathematics, 66, 119-124. 
9. Lima, Asvald (1973), "On simplicial and central measures, and 
split faces", Proceedings of London Math. Society, 26, 707-728. 
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On Nearly Strategic Measures

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On Nearly Strategic Measures

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