A Capacitability Theorem in Measurable Gambling Theory

8/9/2019 A Capacitability Theorem in Measurable Gambling Theory

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A Capacitability Theorem in Measurable Gambling TheoryAuthor(s): A. Maitra, R. Purves, W. SudderthSource: Transactions of the American Mathematical Society, Vol. 333, No. 1 (Sep., 1992), pp.221-249

Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2154107

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TRANSACTIONSOF THEAMERICANMATHEMATICALOCIETYVolume333, Number 1, September1992

A CAPACITABILITYTHEOREM IN MEASURABLEGAMBLING THEORY

A. MAITRA, R. PURVES,AND W. SUDDERTHABSTRACT. A player in a measurablegambling house F defined on a Polishstate space X has available,for each x E X, the collection ?(x) of possibledistributions a for the stochasticprocess xI , x2, ... of future states. If theobject is to control the process so that it will lie in an analytic subset A ofH = X x X x * , then the player's optimal reward is

M(A)(x) = sup{a(A): a E (x)}The operator M(.)(x) is shown to be regular n the sense that

M(A)(x) = infM({T < oo})(x),where the infimum is over Borel stopping times T such that A C It < oo} .A consequence of this regularityproperty s that the value of M(A)(x) is un-changed f, as in thegambling heoryof Dubins and Savage, heplayer s allowedto use nonmeasurable trategies. This last result is seen to hold for bounded,Borel measurablepayoff functions including that of Dubins and Savage.

1. INTRODUCTIONLet X be a nonempty Borel subset of a Polish space and let P(X) be the col-lection of countablyadditive probabilitymeasuresdefined on the Borel subsetsof X. Give P(X) its usual weak topology so that it too has the structureof aBorel subset of a Polish space (see Parthasarathy 16] for information about theweak topology on P(X)). An analytic gambling house F is a mapping whichassigns to each x E X a nonempty subset F(x) of P(X) in such a way thatthe set

F= (x, y) E Xx P(X): ye F(X)}is analytic. Starting at some initial state x E X, a player in the house Fchooses a measurablestrategy a available at x, which means a sequence a =(ao, a1, *.. ), where e0 E F(x) and, for n = 1, an is a universallymeasurablemapping from Xn to P(X) such that an(xI, x2, ... , xn) E F(xn) for every(xl, X2, ... , xn) e Xn . Every measurable strategy a determines a probabilitymeasure,also denoted by a, on the Borel subsets ofH = X x X x

Received by the editors May 16, 1990.1980MathematicsSubjectClassification 1985 Revision). Primary60G40, 93E20, 04A15, 28A12.Key words and phrases. Measurablegambling, stochastic control, regularity,capacity, analyticsets, hyperarithmetic ecursion.The third author'sresearchwas supportedby National Science FoundationGrant DMS-8801085.(D1992 AmericanMathematical ociety0002-9947/92 $1.00 + $.25 per page

221


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222 A. MAITRA,R. PURVES,AND W. SUDDERTHThe probabilitymeasure a can be regardedas the distributionof the coordinateprocess h = (h1, h2, ...), where h1 has distribution a0 and h,+1 has condi-tional distribution an(xI, x2, ... , xn) given hi = xl, h2 = X2, ... , hn = XnFor each x E X, let ?(x) be the collection of measurablestrategiesavailableat x.The optimal reward operator M assigns to each bounded, universallymea-surable function g: H -+ 91 the function Mg defined on X by

(Mg)(x) = suP{( g d: c E (x))}If g is the indicatorfunction of a universallymeasurableset B , we write M(B)for Mg. Thus, for fixed x, M(.)(x) is a set function. Regularitypropertiesof this set function were studied in earlier papers [12, 13, 19] and the majorresult of the present work is another such propertywhich is analogous to thecapacitabilitytheorem of Choquet [2].To state the result,it is convenientto introducetwo topologies on H. Let T1be the product topology on H when X is assigned the topology under whichit is a Borel subset of a Polish space, and let T2 be the producttopology on Hwhen X has the discretetopology. The words "Borel,""analytic,""coanalytic,"and "universallymeasurable,"when used to qualify subsets of H, will refer tothe topology T1 while the words "closed,""open,""clopen," and "G3,"willrefer to T2. For a subset E of H, define the function M*(E) on X by

M*(E)(x) = inf{M(O)(x): E C 0 and 0 is Borel, open}.Our main result is the following theorem.Theorem1.1. If A is an analyticsubsetof H, then M(A) = M*(A).

This theorem was proved in [13] for the special case when X is countableand also for the special case when F(x) is finite for every x. For these cases,it was shown in [13] that M* (.)(x) is a right-continuous(with respect to thetopology T1) capacity for each x. The result then follows from the capac-itability theorem. It was also shown in [13] that M*(.)(x) is not a capacity ingeneral.The proof of Theorem 1.1 will use ideas and techniquesfrom the proof of thecapacitabilitytheorem, but also from gamblingtheory and effective descriptiveset theory. The principaldifficulty n provingTheorem 1.1 is in establishingthemeasurabilityof the function M*(A)(.). It is alreadyhard to do so when Ais a Borel, closed subset of H (see [12] and remarksin [19]). We get aroundthis problem by considering an effective surrogateof the function M*(A)(*)and formulatingan effective refinementof Theorem 1.1. This will involve themachinery of hyperarithmeticrecursion, which will be explained in ?3. Ourproof of the effective refinementof Theorem 1.1will proceedin stages. The firststep will be to establish the result for effective Borel sets which are countableintersections of open sets in the topology T2. This is done in ?5. The nextstep, carriedout in ?6, is to extend the result to effective Borel sets which arecountableunions of countableintersectionsof open sets. The final step, in ?7,proves the result for effective analytic sets (Theorem7.1). Theorem 1.1 is thendeduced in ?8.We regardTheorem 1.1 as a fundamental approximationresult in gamblingtheory as is the usual capacitabilitytheoremin probabilityand measuretheory.


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A CAPACITABILITYHEOREM N MEASURABLEGAMBLINGTHEORY 223It will be applied in ?9 to prove Theorem 1.2, which answers an old questionabout the adequacy of measurable strategies.Recall that the gambling theory of Dubins and Savage [4] takes place in avery general finitely additive framework in which a player is not restricted tomeasurable strategies. In the theory of Dubins and Savage, as extended byPurves and Sudderth [18], the probability a(A) is defined for every analyticallymeasurable (that is, measurable with respect to the a-field generated by analyticsets) set A under every strategy ac, measurable or not. In consequence, f g dais defined for every bounded g: H -* 91 which is upperanalytic in the sensethat {h E H: g(h) > r} is analytic for every real r . For such a payoff functiong, the optimal reward to a player with initial state x who is not restricted tomeasurable a is

(Fg)(x) = sup {Jgda: a available at x}.However, the player gains no advantage through the use of nonmeasurable a's.Theorem 1.2. If g is boundedand upperanalytic, then rg = Mg.

This theorem generalizes Theorem 2.1 of Dubins et al. [5] which correspondsto the special case when g(h) = lim u(h,) for some bounded, upper analyticu: X --R 1.2. DEFINITIONS AND NOTATION

N will denote the set of positive integers, w the set of natural numbers, and91 the set of real numbers. If s, t are finite sequences of natural numbers, wewrite IsI for the length of s, si for the ith coordinate of s for 1 < i < IsI,s D t to denote that s extends t, and st for the catenation of s followed byt. If a E oN and n E N, the finite sequence (a(l), a(2), ...a, ((n)) will bedenoted by cZ(n).We use X exclusively to denote the state space. It will always be a Borel sub-set of a Polish space. Forp EXm and (xl, x2, ... , xn)eXn, p(x1, X2, .. ., xn)or pxl, x2 ... xn will denote the element of Xm+n obtained by catenating pand (xl, x2, ... , xn). The symbol H will be reservedfor XN, the space ofhistories. For h E H and i E N, hi will denote the ith coordinate of h. Ifh, h' e H and n e o, we write h nh' if hi = h for i = 1,2,...,n.If p e Xm and h E H, ph will be the element of H obtained by catenatingp and h. For k e N, Pk will denote the function on H to Xk defined bypk(h) = (h1, h2, ... , hk). If B C H and p E Xm, then Bp will denote theset of h E H such that ph E B. Similarly, BPk will be the set-valued functionon H whose value at h is the set Bpk(h).A mapping T from H to N u {oo} is called a stoppingtime if

'r(h) = n E N and h'- nh imply T(h') = n.A stop rule is an everywhere finite stopping time. A stopping time T is Borel(universally)measurableif for each n E N, the set fT < n} is Borel (uni-versally) measurable in H. If T is a stopping time, then h, and p, arefunctions on the set fT < OO} whose values at h are hT(h) and PT(h)(h) =(hl, h2, ... , hT(h)), respectively. Similarly, Bp, is the set-valued function on


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224 A. MAITRA, . PURVES, NDW. SUDDERTH{r < oo} whose value at h is Bp,(h). If x E X, define r[x] on H by

T[x](h) = T(xh) - 1.Then r[x] is again a stopping time or identically equal to zero. We writer[(Xl x2 ... Sxn)] or just T[x1, x2, ... ,xn] for T[x1][x2]2 [xn]. If k E N,define the stop rule T Ak by

(TA k)(h) = minimum{T(h), k}.Let t be a stop rule. We say that a function q on H to a set Y is determinedby time t if

t(h) = k&h kh' imply 0(h) = q(h').A subset K of H is said to be determinedby time t just in case the indicatorfunction of K is determinedby time t. It is easy to prove that a subset of His clopen in the topology T2 iff it is determined by time t for some stop rulet. A function V/ on H x N to Y is adapted if

h kh' implies yV(h,k) = yV(h',k).Suppose a is a measurablestrategyavailable at x and p E Xm We define ameasurablestrategy a4[p]available at 1(p), the last coordinate of p, as follows:

(UI[P])0- (m(P)and, for n > 1,(a[p])n(Xl , X2, ... , xn) = lm+nn(PX1 , X2 Xn) .

It is easy to verify that the measures (on H induced by) c[p], p E Xm,are a version of the conditional ay-distributionof (hm+? hm+2, ...) given(hl, h2, ... , hm) = p. If T is a universallymeasurablestopping time, a4p,] isa mappingon ft < oo} whose value at h is a4[p,(h)]By virtue of Lemma 2.2 in [11], we can fix a Borel measurable mappingv: P(H) x X -- P(H) such that for each ,u E P(H) and x E X, v(u, x) isa regular conditional ,u-distribution of (h2, h3, ...) given h1 = x. For ,u EP(H) and x E X, we write ,u[x] for v(,u, x); more generally, f xl, x2, ....

Xn E X, we shall write A (xi, x2, ... ., xn)] or Au[x1,x2, ... , xn] for 1u[x1][x2].*. [xn] . If ,u E P(H), /uo will denote the marginal distribution of ,u on the firstcoordinate of H. If T is a universallymeasurablestopping time, ,4pT] willdenote the mappingfrom fT < oo} to P(H) whose value at h is i4[p, h)]. Weremark that this notation is in agreementwith that of the previous paragraphin the following sense. If a is a measurablestrategyand ,u the probabilitymeasure induced on H by ao, then, for almost all (,u)p E xm, the measureinduced by the strategy a4[p]agreeswith ,u[p]Suppose Tnare universallymeasurablestopping times such that Trn Tn+1on {fTn+l< oo}, n E N. Assume that, for each n E N, On: H x N * P(H)is universally measurableand adapted. Let AE P(H). Define, for each Borelsubset B of X,

qo(B) = AO(B)


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A CAPACITABILITY HEOREM N MEASURABLEGAMBLINGTHEORY 225and, for n > 1,

qn(BIlx, X2, ..., xn) = ([x1, x2, ...,n)(B)if n


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226 A. MAITRA,R. PURVES,AND W. SUDDERTHSuppose now that Z1 and Z2 are Al-recursivelypresentedcompact metricspaces. Then Z1 x Z2 and P(Z1), the set of probability measures on theBorel subsets of Z1, are again Al-recursivelypresentedcompact metric spaces(Louveau[7, 9]).In what follows, our terminology and notation, pertaining to concepts ineffectivedescriptiveset theory, are takenfrom Moschovakis[15].Following Louveau [8, p. 13], we say that the pair (W, C) is a coding ofBorel subsets of a Al recursivelypresentedcompact metric space Y if(a) W is a HI1subset of w x ;(b) C is a H11 ubset of cw x cox Y;(c) the set {(a, n, y) e C x cox Y: (a, n) E W&(a, n, y) ? C} is 1;(d) for fixed (a, n) E Wcw c, the section Ca,n = {y E Y: (a, n, y) E C}is Al(a);(e) If P C Y is Al(a), then there is an n such that (a, n) E W andP = Ca,n

The rest of this section will be devoted to proving an effective separationtheorem which will be needed in the sequel. In what follows and until furthernotice, X will be 2a', the space of sequences of O'sand l's, and H will beX x X x * - .Fix a coding (W, C) of Borel subsets of the Al-recursivelypresentedcom-pact metric space H. We will be interested in (a, n) E W such that Ca,,, isBorel and open. Define W C wc) x wt)as follows:(a, n) E W*- (a, n) E W

&(Vh)[(a, n, h) ? C or (3m)(Vh')(h'm_ h -* (a, n, h') E C)].Clearly, W is HIl,and if (a, n) e W, then (a, n) E W if and only if Ca,,nis open. Set C=Cfn(WxH).Then

(i) C is HI;(ii) the set {(a, n, h) e o x cox H: (a, n) E W&(a, n, h) ? C} is H1;(iii) P c H is Al(a) and open if and only if P = Ca,, for some (a, n) EW.Lemma 3.1. If A and B are 1l subsets of H such that A can be separatedfrom B by an openset, then there is a Al-recursive toppingtime T such that

A C {T< 00} C BC.Proof. Define P C H x N as follows:

(h, n) EP -*(Vh')(h'=nh -h' ? B).Then P is Il and (Vh E A)(3Jn)((h,n) e P). So, by the Kreisel selectiontheorem [15, p. 203], there is a Al-recursive function f: H -- N such that(Vhe A)((h, f(h)) e P) . Let

Q = {(h, n) e H x N: (3h'E A)(h' nh &f(h') = n)}.


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A CAPACITABILITYHEOREM N MEASURABLEGAMBLINGTHEORY 227It follows that Q is X1 and Q C P. Next, we define S C N x co as follows:

(n, m) E S +-- x*, m) E W&(Vh)(Vh')(h E Cx*, m&h nh' h' E Cx*,m)&(Vh)(h E Cx*,m (h, n) E P)&(Vh)((h, n) E Q h ECx*,m)

where x* is a fixed Al point in co'. Then S is FIT Use the first principleof separation for X1 sets [15, p. 204] to see that (Vn E N)(3m)((n, m) ES). Invoke the Kreisel selection theorem one more time to get a Al-recursivefunction g: N -- c such that (VnE N)((n, g(n)) E S). Finally, define z onH as follows:h) least n such that h

e Cx*g(n) if (3n E N)(h E Cx*,g(n))oo otherwise.

Then it is easy to verify that T satisfiesthe assertionof the lemma. 5As immediate consequencesof the lemma, we have the following two corol-laries.

Corollary 3.2. If V is a Al, open subset of H, then there is a Al recursivestoppingtime T such that V = {r < oo}.Corollary3.3. If K is a Al, clopen subset of H, then there is a Al-recursivestop rule T such that K is determinedby time Tr.

Classical (boldface) versions of the above resultswere proved in [10].4. A FUNDAMENTAL INEQUALITY

Let X = 2w, H = Xx Xx , and I C Xx P(H) be aglobal gamblinghouse which is a El subset of the Al-recursivelypresented compact metricspace X x P(H).Let E C H be universallymeasurableand let x E X. We can define(4.1) M(E)(x) = sup{ i(E): P E Y-(X)}, x E X.There should be no confusion with the symbol M introduced in ?1, whereX(x) denoted the set of measurablestrategiesavailable at x. Here we have no"local"gamblinghouse F, but in case I above had been induced by a "local"gamblinghouse, the M of (4.1) would have coincided with the same symbolof ?1.Next we define the effective surrogate M of the function M* of ?1. How-ever, M has to be defined simultaneouslyfor a X1 subset of H and all itssections. So let E be a X1 subset of H and let xl, x2, ... ,Xk, x E 2w. Wedefine(4.2) M(E; x1, X2, ..., Xk; x) = infsup{,u(O): ,u E E(x)},where the inf is takenover all Al((xl, x2, ... , Xk, x)), open sets 0 containingEx1x2 Xk. Here (a I, a2, ... , ak) is the member of cow which codes thefinite sequence a I, a2, ... , ak of members of cow, as described in [15, p.40]. Since 2w C cow, the coding can be used for finite sequences of members


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228 A. MAITRA,R. PURVES,AND W. SUDDERTHof 20 as well. We allow the finite sequence xl, x2, ... , xk to be the emptysequence in (4.2). In the sequel, the left side of (4.2) will be abbreviatedbyM(ExIx2 Xk)(X). Similarly,if a E cWWand F is a 11 a) subset of H, wedefinethe relativization Ma of M as follows:(4.3) Ma(F; xl , x2 , ..., .xk; X) = inf sup{,u(0): ,u E I(X)},where now the inf is taken over all Al ((a, xl, x2, ..., Xk, x)), open sets 0containing Fxlx2 ... xk . The left side of (4.3) will be abbreviatedby

Ma(Fxlx2.Xk)(X).Now suppose that F = Ex,x2 Xk for some X1 subset E of H andxI, X2, ..,Xk E 20. Set a = (xl, x2, ...,Xk) . Then, as is easy to verify,

(4.4) Ha(F; ; x) = M(E; xl, X2, ***,~ Xk; X)We will use (4.4) without explicit mention in the sequel.Lemma 4.1. If E C H is X1 then the set

{(h, k, x, a) E H x N x X x [O, 1]: M(Ehlh2 hk)(x) > a}is X1.Proof.M(Ehlh2...hk)(x)>a

(V)(r)(Vn)[r < a& ((hi, h2, . ,hk, x), n) E W& Ehlh2 hk C C(h1,h2,...,hk,x),n

(3Yu (X)) (/C(h,, h2hk, x), n) > r)],where r runs throughthe rationals in [0, 1], and we think of 20 as as beingembeddedas a 1I7 subsetof coW One sees easilyby imitatingthe computationin the proof of Theorem 2.2.3(a) in Kechris [6] that the relationabove is ?1 . 1Lemma4.2. If E C H is X1 and T is a Al-recursive topping time on H, thenthe set {(h, a) E Hx [O, 1]: T(h)< oo&M(EpT)(hT)>}is XI.Proof.

r(h) < oo&M(EpT)(hT) > a+-+3k E N)[r(h) = k&M(EhIh2 .hk)(hk) > a].The above condition is XVby virtue of Lemma 4.1. oLemma 4.3. Let E C H be 1 and r a Al-recursivestopping time on H.Fix xo E X and e > 0 rational. Then there is a Al(xo)-recursive unctiong: H- [O, 1] such that

(i) r(h) < ox =T g(h) > M(EPTO)(h)(ii) r(h) = k < oo&h kh' = g(h) = g(h'), and(iii) SUP/1EL(X0)g(h)du(h) < SUP,LEL(x0)T


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A CAPACITABILITYHEOREM N MEASURABLEGAMBLINGTHEORY 229Proof. Let X* = Um,,coXm. Now X* can be endowed with a Al-recursivepresentation so it becomes a Al-recursively presented Polish space. Define: H -4X* as follows:

VIh) I(hi ,h2 h.,h) if T(h)


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230 A. MAITRA,R. PURVES,AND W. SUDDERTHThen P is FIl(xo) and one verifieseasily that (Vh)[ (h)< oo - (3n)((h, n) EP)]. An application of Kreisel's selection theorem yields a Al(xo)-recursivefunction f: H -* w such that (h, f(h)) e P whenever -r(h) M(E)(xo) - e/3.

Nowv(O) v[pj](Op,)(h,)dv(h) .

If r(h)


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A CAPACITABILITY HEOREM N MEASURABLEGAMBLINGTHEORY 231Theorem 5.1. Let X = 2w, H = 20 x 20 x *,and I c X x P(H) a 11, globalgambling house. Let Vn be open subsets of H such that Vn2 Vn+1, n E N,and assume that the relation h E Vn is Al in N x H. Set G = nnEN Vn ThenM(G) = M(G).Proof. By Corollary3.2, for each n E N, there is a Al-recursivestoppingtimeTn suchthat Vn = {Tn < X}. Wemayassumewithout oss of generalityhatTn < Tn+1 on Vn+lforall n E N.Fix xo E X and e > 0. It will sufficeto prove that M(G)(xo) > M(G)(xo) -e. We will do this by defining ,u E I(xo) such that(5.1) /u(G) > M(G)(xo) - 8The definition of ,u will involve repeated applicationsof Theorem4.4.To startwith, use Theorem 4.4 to choose AE I(xo) such that(5.2) M(GpTI (hTj dA(h) > M(G)(xo) - 8/2.Next, define f: H x N x H'-- [0, 1] as follows:

M(Gh1h2. hmPT+i[hi , h2,... hm](h'))(hTn+i[hi ,h2,... hm])(5.3) f (h, m , h') =if m < rn+l (hlh2 hm)h') < x,

0 otherwise.Using Lemma 4.1, one verifies that f is an upper analytic function. It thenfollows from a selection theorem [12, Lemma 2.1] (see also [1, Proposition7.50]) that there is an analyticallymeasurablefunction q/: H x N P(H)such that qi(h, m) E 1(hm) and

f(h, m, h') yi(h, m)(dh') > sup f(h, m, h') v(dh') -vEl(hm) 2n+1for every h E H and m E N. The function q/ may not be adapted,but this iseasily set right by defining On: H x N P(H) as follows:

Obn(h, m) = V((hi, h2 p.. ,hm x* , x*, ...), m),where x* is an arbitrary ixedpoint in X. We then have(5.4) Jf(h, m, h')On$(h, m)(dh') > suip f(h,J m, h') v (dh') - 2n+forall hEH and mEN.It now follows from (5.2), (5.3), and (5.4) that, if rn+1h) > mi, then

M(Gh1 h2 * hmPzn+l [hi, h2,... hm] h'))Tn+1hi, h2,.. hm] M(Gh1h2... hm)(hm) -/2n+1


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232 A. MAITRA,R. PURVES,AND W. SUDDERTHwhere the last inequality is obtained by an application of a relativized ver-sion of Theorem 4.4 to the AI((hI, h2, ..., h,)) set GhIh2. hmand theAl (hi, h2, ... , hm))-recursive stopping time r,+I[hl, h2, ... , h,,]. Hence,for any h such that T,r(h)< 0, we can set m = Tz(h) in (5.5) to obtain

T(n+iUNn(h)]


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A CAPACITABILITYHEOREM N MEASURABLEGAMBLINGTHEORY 233Lemma 6.1. If (En) is a S1 sequenceof subsets of H and M(En)(x) = 0 foreach n E N, then M(UfENEn)(x) = 0.Proof. Let e > 0 be rational. Define P C N x co by

(n, m) EP +(x, m) EW&En C Cx,m&(V i ? (x)) (Cx,m) < 2n)Then P is Ill(x), and the hypothesisimplies that (VnE N)(3m)[(n, m) E P1.So, by Kreisel'sselection theorem, there is a Al(x)-recursivefunction f: N -*wc such that (VnE N)[(n, f(n)) E P]. Let

hE O+-+ (3n E N)[h ECx,f(n)]Then 0 is a Al (x), open set containing UnENEn. Hence

M (U En) (x) 0 rational. Define a stoppingtime zT as follows:r least k E N such that M(Ehi h2 hk)(hk)> 1-8,

Tc(h)= 4if (3k E N)[M(Eh1h2 ... hk)(hk) > -ox otherwise.Lemma 6.3. (a) The set {(h, n) e H x N: Te(h) < n} is Xl.(b) The set {IT < x} is 1l in H.Proof. The relation M(EhIh2 ... hk)(hk)> 1 - e in h, k can be rewrittenas

(]i E Y-(hk))( (Ehl h2 . . hk) > 1 - e).This is clearly El in H x N. The assertion in (a) follows immediately fromthis and (b) follows from (a). OLemma6.4. Let E and T, be as above. Then M(E n {IT = occ})(x) = 0 forevery x E X.Proof. Towardsa contradiction,assume there is an x such that

M(E n{e = oo})(x) > 0.So there exists , E I(x) such that g(E nT r = 0}) > 0. By the Levy zero-onelaw, as n -* oo,(6.1) u[Pn(h)][(E {IT = oc})Pn(h)] -- (E n IT, = oo})(h) a.s. (g),where the indicator function of a set C is being donated by the same sym-bol. Denote by Cl the subset of H where convergenceholds in (6.1), so that4u(CI)= 1. Let C2 be the set of h E H such that 4[Pn(h)]E 1(hn) for every


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234 A. MAITRA,R. PURVES,AND W. SUDDERTHn E N. Since I is a global gambling house, it follows that ju(C2)= 1 . Conse-quently, lu(C1n C2 n E n IT, = oo}) > 0. Pick h* in C n C2fnE nIT{ = oo}.So, by (6.1), there is m > 1 such that

4U[Pm(h*)][(En {Te = w})Pm(h*)] > 1- .This implies that A[pm(h*)](Epm(h*)) > 1 - C, so M(Epm(h*))(h*) > 1 - C,since lu{pm(h*)]E 1(h*). But then T6(h*) < m, contradicting the fact thatTe(h*) = 00. This completes the proof. ULemma 6.5. M({T, < xo})(x) < M(E n {IT < oo})(x) + 2e for every x E X.Proof. We write T for T. Choose A E ?(x) such that

A(T{ < 00}) > M({T < oo})(x) - e/2.Next, use a selection theorem [12, Lemma 2.1 or 1, Proposition 7.501to find,for each n E N, an analytically measurable selector V/n:H -* P(H) for the set{(h, 4u)E H x P(H): ,uE X(hn)} such that

yin(h)(EPn(h))> M(Epn(h))(hn) - e/2for every h e H.Let x* be a fixed element of X. By the von-neumann selection theorem[15, 4E.9], fix an analyticallymeasurable selector f*: H P(H) for the set1. Define

k1(h, k) = V/k((hl, h2, ..., hk, X*, X*, ))and, for n > 2,f*(hk) if I 2, Tn = Ti + n - 1. Then it is straightforward overify that A, q$n and Tn satisfy the assumptionsof (b) in the definition of aglobalgamblinghouse. Let ,u be the sequentialcomposition of (A, On, TO), SOu e ?(x).Hence,

M(E n fT < oo})(X) ? u(En fT < X0}) = jpPT(EpT) d){T (1 - 3/2))({T < oo}) > M({T < oo})(x) - 2C.

This completes the proof. ElWe now give the main resultof this section.Theorem6.6. Let X = 2w, H = 2w x 20 x ,and I C X x P(H) be a X1,globalgamblinghouse. Supposethat the relation h EBn, m is Al in H x N x Nand that Bn,m is open in H for every n, m E N. Let En = nmENBn,m,n E N, and E = UnENEn. Then M(E) = M(E).Proof. Fix xo E X and e > 0 rational. Consider the stopping time Te as-sociated with the set E. By Lemma 6.3, {T, < oo} is a X1 set. So, by a


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A CAPACITABILITY HEOREM N MEASURABLEGAMBLINGTHEORY 235relativized version of Lemma 4.3 in [12], there is a Al (xo) set D in H suchthat {z < oo} C D and

M(D)(xo) < M({Te < oc})(x0) + e.Now the El set {IT < oo} can be separated from the Al (xo) set Dc by an openset. Hence, by a relativized version of Lemma 3.1, there is a Al (xo)-recursivestopping time t such that {IT < oo} {t < oo} C D. Set O={t < oo} . Then(6.2) M(O)(xo) < M({rE < oo})(xo) + e< M(E n{Te < oo})(xo) + ? M(E nO)(xo) + 3C,where the second inequality is by virtue of Lemma 6.5. On the other hand,

EnO =[U lnBn,m] nfn[{t?k}] u n n[Bn, mn{t>k}].LnEN mEN J kEN nEN mEN kENMoreover, by virtue of Lemma 6.4,

M(E n OC)(xo) < M(E nT r = oo})(xo) = 0.This fact together with the representation of E nOc above enables us to invokea relativized version of Lemma 6.2 and conclude that(6.3) MXOE n Oc)(xo) = 0.Now use (6.3) to obtain a Al (xo), open set O' containing E n Oc and suchthat(6.4) M(O')(xO) < e.6Then 0 U 0' is a Al (xo), open set and contains E. Hence,

M(E)(xo) < M(O UO')(xo) < M(O)(xo) + M(O')(xo)< M(E n o)(xo) + 4e < M(E)(xo) + 48,where the third inequality is by virtue of (6.2) and (6.4). Since e is an arbitraryrational, this proves that M(E)(xo) < M(E)(xo). As the inequality in theopposite direction is obvious, thlis completes the proof. n1The next result is the raison d'etre for this intermediate step in the proof ofthe effective refinement of Theorem 1.1.Corollary6.7. Let (An) be a Vl sequence of subsetsof H such that AnC An+,n E N. Let A = Un NAn. Then M(A) = limM(An).Proof. Fix x0 E X. Plainly, M(A)(xo) > limM(An)(x6). It remains only toprove the inequality in the opposite direction. Let r be a rational in [0, 1]such that M(A)(xo) > r. We need to show that limM(An)(x0) > r. Supposenot. Then, for each n E N, M(A,)(xo) < r. Define P C N x a) as follows:(n, m) E P +-* (xo, m) E W&A, C Cx0,m &((VP E (Xo))(Ii(Cxo,m) < r)Plainly, P is 11 xo) and (Vn E N)(3m)[(n, m) E P] . So, by Kreisel's selectiontheorem, there is a Al(xo)-recursive function f: N -+ co such that (Vn EN)[(n, f(n)) E P]. Let

h EB *- (3k E N)(Vn > k)[h E Cxo,f(n)]


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236 A. MAITRA,R. PURVES,AND W. SUDDERTHThen A C B, and by Theorem 6.6 (relativizedto xo) , Mx0(B)(xo) = M(B)(xo) .Hence,

M(A)(xo) < Mxo(B)(xo)= M(B)(xo) = supM ( n CX0,f(n) (XO)k \n>k< supM(Cx0,f(k))(xo) < rk

which yields the desired contradiction. El7. AN EFFECTIVE CAPACITABILITY THEOREM

For fixed x E X, the monotonicity propertyand the propertystatedin Corol-lary 6.7 suggest that the set function M(.)(x) may be an "effective"capacityinthe sense of Louveau [9]. However,this is not so, as the "going down" propertyalong decreasing sequences of TI compacts fails to hold in general (see [13]).In consequence, the "effective" capacitability theorem of Louveau cannot beapplied directly, as Choquet's capacitabilitytheorem was in some special cases(see [13, Theorem 3.6]), to prove Theorem 1.1. Nonetheless, we will now com-bine the proof of the abstract capacitabilitytheorem -or, to more preciselyidentify the provenanceof these ideas, Sierpifiski'sproof that an analyticset ismeasurable[21, pp. 48-50]-with the proof of Theorem 5.1 to producea proofof the effective refinementof Theorem 1.1. But firstwe must explainsome morenotation from the effective theory.Let U: co"x H co be a Ill-recursivepartialfunction such that whenevera E coZ andf: H co is a HI a)-recursivepartialfunction, there is a Al a)point 8 E w' such that if f(h) is defined,then U(,B, h) is defined and f(h) =U(,f, h). Following Kleene, we shall denote the partial function U(,f, *) by{f,}. We write {fl}(h) 1 to mean that {f,} is defined at h. Next, fix a HI-recursivepartial function d: cowx co * co" such that d(a, *) enumeratestheAI(a) points of cow' The existence of U is establishedin [15, 7A.1] and thatof d is proved in [15, 4D.2].Here at last is the effective refinementof Theorem 1.1.Theorem 7.1. Let X = 2wo, H = 20 x 20 x .,and X C X x P(H) be a II,global gambling house. If A C H is a 1Iset, then M(A) = M(A).Proof. We use [15, 4A.1] to write A as follows:(7.1) h EA +-t (3a E coN)(Vn E N)[h EB(z(n))],where h E B(s) is a Al relation in UnEw(onx H, B(s) is clopen in H foreach fixed s E UnElcon, and s D t implies B(s) c B(t). Define(7.2) h E A(s) *-+ (3m)[IsI = m & (il < sl)(3i2 < 52) ... (3im < sm)(3a)(7.2) (Vn e N)(h E B((il, i2, ... , im)Zi(n)))].

Plainly, the relation h E A(s) is 1I in UnEWon x H.Fix xo E X and e > 0. We will construct ,u E X(xo) such that(7.3) /1(A)> M(A)(xo) - e.

The definitionof ,u dependson an inductive constructionof various objects.The conditions that these objects have to satisfy are stated in the next lemma.


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A CAPACITABILITY HEOREM N MEASURABLEGAMBLINGTHEORY 237Lemma 7.2. Let xoE X, e >O, A E Y(xo), m E C, m 1:H -+, K, aclopensubsetof H, rn a stop rule on H, and On:H x N -* P(H) such that(a) for every n E N, mn+. Kn, TZI,and '/ are universally measurable,Kn+jc Kn, Tn< Tn+1, Kn, and m*+i are determinedby time Tn, On(h, k) EX(hk) for every k E N, and q$n s adapted;(b) h E Kn iff h E B((ii, i2 , ..., in)) for some i1 < mt, i2 < m*(h),in M(A)(xo) - e/2; and(d)forevery neN and heH,| M(A((m*, m* (h) , ,m*+, (h)))qn (h)PTn+1[qn,(h)])K+lqn(h)

(hTn1qn^]On(hgTn(h))(dh)> M(A((m*, m*(h), ... , m*(h)))qn(h))(hTn)-2n+l

where qn(h)= pTn(h), n e N.Let ,u be the sequential compositionof (A, tOn Tn) . Then(i) nnlENKn c A, and(ii) f(nneN Kn) > M(A)(xo) 8.

Proof. To prove (i), let h E nfeN Kn and defineS= {s E U cn: (3k)[IsI= k &sl < m*&s2< m*(h)nEco

&... & Sk 1 as follows:

Y (h) =-{ M(A(mt, m*(h), ... , m (h))qn h))(hTn) if h E Kn,otherwise.In what follows, expectations and conditional expectations are with respect tothe probabilitymeasure ,u on H. Accordingto (c), we have

E(Y1) > M(A)(xo) - 8/2,while (d) yields for almost all (,u) h E H,

E(Yn+l PTJ) > M(A(m* , m*(h), ... , m*(h))qn(h))(hTn) - 2n+1'wherethe conditional expectation is with respectto the pre-Tn a-field. Hence

E (Yn+ PTJ >Yn - /2n+lso that

E(Yn+i) > E(Yn) - e/2n+'.


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238 A. MAITRA, R. PURVES, AND W. SUDDERTHConsequently,

E(Y,,,,) > E(Yj) - 2+ +***+2+)for every n > O, and so(7.4) lim sup E(Yn) > M(A)(xo)-e.en

Now for each n E N, {Yn > O} C Kn. Since 0 < Yn< 1, it follows thatu(Kn)> E(Yn) for every n E N. Hence,(F Kn) = lim (Kn)> limsupE(Yn) > M(A)(xo)

wherethe firstinequalityis byvirtue of (7.4). This establishes(ii) and completesthe proof of the lemma. OWe returnto the proof of Theorem 7.1. Since I is a globalgambling house,,u, constructed in Lemma 7.2, belongs to X(xo). So (7.3) is an immediateconsequence of Lemma 7.2. It therefore remains to show that the hypothesesof Lemma 7.2 are satisfied. We begin with a lemma.Lemma7.3. Forevery n > 2, thereis a Al-recursiveunction gn: H x (onx NCcoWuch that if gn(h m,I , ..., mn, k) = ag,then a is a AI((hl, h2, ..., hk)) point and {a} is a A1((hl, h2, hk))-recursive top rule T on H such that the Al((hI, h2, ... , hk)), clopenset

mI m2 mnU U U B((il, i2, - , in))hlh2 ...hki1=0 i2=O in=Oisdeterminedbytime T. Furthermore, if h=kh', then gn(h i1, m2, ..., mn, k)= gn(h', ml, M2 ... , mn, k).Proof. Define P C H x oinx N x woas follows:(h,m1, m2, ... Mn k, j) E P

d((hj, h2, hk), j) J&(Vh')[{d((hl, h2, ..., hk), j)}(h') t&d((hj, h2,..., hk), j)}(h') > 1]

&(Vh1)(Vh1)(Vi)[{d(h, h2,.. ., hk), j)}(h') = i&h' ih"{d((h1 , h2, hk), j)}(h") = i]

&(Vh')(Vi) [d((hl h2,.*-,hk), j)}(h') = imI1 m2 mn

((Vh) hl =ih' -y h"lE U U... U B((ii, i2,... in))hlh2 ... hkil =0 i2 =? in =O

ml m2 Mnv(Vhl() hi h'h" hllU U 40U B((i1, i2, in))hlh2 ... hk

il =0 i2-? in=O

Then P is III and an applicationof a relativized version of Corollary3.3 willshow that(Vh)(Vm)(Vm2)... (Vmn)(Vk)(3j)[(h, mi1, M2, ..., mn, k, j) E P].


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A CAPACITABILITY HEOREM N MEASURABLEGAMBLINGTHEORY 239So, by Kreisel's selectiontheorem,there is a Al-recursivefunction f: H x WnxN -- w such that

(Vh)(VmI)(Vm2) . (Vmn)(Vk)[(h,~Ml M2, Mn. , k, f(h,MI M 2, ..,Mn) k)) E P].Finally, setgn(h Ml, m2 ..,mn ,k)L= d((hl, h2, ...,~ hk) f(h, Ml M2, Mn.m k)),

where h=(hi, h2, .. , hk, x*, x*, ... ) and x* is a fixed Al point in X. Itis now easy to verify that gn satisfies the assertions of the lemma. 51We are now ready to complete the proof of Theorem 7.1. Observe, first,that(A(M))m is a I7 sequence, A(m) C A(m+ 1), and UmEwA(m) = A. Hence,

by Corollary 6.7, there is mt such that(7.5) M(A((min)))(xo)> M(A)(xo) - E/4.Define K1 as follows:

h E K1 +-* (]i < mi)[h E B((i))J,so K1 is a Al, clopen set and A((mi)) c K1 By Corollary 3.3, there is a Al-recursive stop rule z1 such that K1 is determinedby time T . Apply Corollary4.5 to the Il set A((mr)) and Al-recursivestop rule Ti at the point x0 to getAE X(xo) such that(7.6) JM(A((mt))PTl)(hTl) dA(h)> M(A((mt)))(xo) - e/4.Since A(mt) C K1 and K1 is determined by time zi, combining (7.5) and(7.6), we get

J M(A((m*))pT1 )(hTr1)A(h) > M(A)(xo) - e/2.Suppose, next, that K1, K2, ..., Kn, Z1, 2, ... Tn, M*, M and(1, 4b2 O-- q$n-I have been defined. Define J: H x Zn x N o+ as follows:J(h, MI, M2, ..., mn, k) = least m such that(7.7) M(A((ml, M2, ..., mn, m))hlh2 ... hk)(hk)

> M(A((m, m2, . . ., mn))hlh2 * hk)(hk) - 2n+2Then J is well defined. To see this, first observe that

(A((m1 ,M2, ... , mn, m))hlh2... hk)mEcois an increasing II ((h1, h2, ... , hk)) sequence whose union is

A((ml, M2, . ,mn))hl h2 ... hkNow use a relativizedversion of Corollary6.7 to see that there exists m satis-fying (7.7).It follows easily from Lemma4.1 that J is analyticallymeasurable.Further-more,h kh' implies J(h, ml, M2, ..., m,k) = J(h', mI, m2, ..., m,k).


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240 A. MAITRA,R. PURVES,AND W. SUDDERTHNow set(7.8) M*n+1(h)J(h m,m M*(h), ., m* h) T,(h)), h E H,so that mn+1 is universally measurableand determined by time Tn. Next,define Kn+1by the formula

h E Kn+1 +-* (3m2)(3m3) ...(3Mn+l)[m*(h) = m2 &m*(h) = M3& *& m*+1 (h)

= mn+1 &(3ij < m*)(3i2 < m2) ... (3in+l < Mn+l)(h E B((ij, i2 *** in+l))]-

It is easily verified, using the facts that the functions m*, m*, ..., mn+ aredetermined by time Tnand that B(s) is clopen in H, that Kn+1 is clopen.Furthermore,Kn+i s universallymeasurable,since the functions m*, m,...,M*+l are and the sets B(s) are Borel. Plainly, condition (b) is fulfilled andn+lFix a function gn+i: H x Wn+l x N -? WW having the properties stated inLemma 7.3. Define Tn+l: H -? N by the formula(7.9) Tn+l(h)= Tn(h)+ {gn+i(h, m*, m*(h), * , m n(h), zn(h))}(h'),where h' = (hTn(h)+l hT,(h)+2 .**). Using the inductive hypothesis and Lemma7.3, it is not hard to verify that Tn+1 s a universallymeasurablestop rule suchthat Tn< Tn+, and Kn+1 is determined by time Tn+1In order to define On proceed as follows. First, define 0: H sup J (h iM,I 2, Mn1..., +, k, h')zv(dh')- 2n+2vEY,(hk) 2+Now an applicationof a relativized version of Corollary4.5 to the

((hA h2, ,hk))


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A CAPACITABILITYHEOREM N MEASURABLEGAMBLINGTHEORY 241set A((mi, M2, .. , n+l))hlh2 hk and the Al((hi, h2, ... , hk))-recursivestop rule {g,+I(h, ml, M2, ...M, , k)} yields

M(A(ml, M2, ..., Mn+l)hlh2 hk)(hk)(7.12) < sup J0(h ,Mml M2 5 ..., 5Mn+1 k,5 h')v(dh')

vEY?(hk)

for every (h, mI, M2, ..., Mn+i, k) E H xWfn+lx N.Define O5,n: x N -+ P(H) by settingVlW((hl,h2, *. ., hk, X*, X*,...)

(7-13) Otn(hk) = *5ml,m*(h),*, m*+,(h),k) if k > Tn(h)Ti7(hk) if k< Tn(h)

where 7: X -+ P(H) is an analyticallymeasurable selector for I regardedasa subset of X x P(H), and x* is a fixed point in X.Now it is easy to check that qAn s universally measurable, adapted, andq$n(h,k) EX(hk) for every h E H and k E N.It follows from (7.9)-(7.13) and the property of q mentioned immediatelyafter (7.10) that for any h E H,JM(A((ml* m2(h) m** n+lh)))qn(h)PTn+1qn(h)](h'))

(hlnl q()])n( h Tn(h))(dhl)J M(A((m* m *(h), m, +(h)))qn(h))(hTn)> M(A((m*, m2(h), ..., m*(h)))qn(h))(hTn) 2n+1

where the last inequality is by virtue of (7.7) and (7.8). Condition (d) of Lemma7.2 now follows from the previous inequality by observingthatA((ml*, m2*(h) ., m*+l (h))qn(h)

is contained inKn+Iqn h) and thatKn+I n h) is determined by time Tn+Iqnh)].This completes the proof of Theorem 7.1. o

8. PROOF OF THEOREM 1.1It is now an easy matter to deduce Theorem 1.1 from Theorem 7.1. So letus return to the setting of ?1. First, assume that X = 2w, F is an analyticgamblinghouse on X, and A is an analytic subset of H = X x X x . . Foreach x E X, let X(x) denote the set of probabilitymeasures on the Borelsets of H induced by universallymeasurablestrategiesavailable at x. It wasobservedby Dellacherie[3] (see also Sudderth[23]) that I is an analyticsubsetof X x P(H) . Furthermore,I is a global gamblinghouse of the type that wasconsideredin ??3-7. By a resultin [15, p. 160], we can choose a E WOWindeed,a can be chosen in 20) such that both A and ? are 11a) sets. Since theglobal gamblinghouse E is X1 a), we modify the set function M introduced


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242 A. MAITRA,R. PURVES,AND W. SUDDERTHin (4.2). Define a new set function M as follows:

M(E; xl ax2 , ... ., xk; x)(8.1) = inf{M(O)(x): 0 is a AI((, x, X2,-, Xk, X)),

open set containing ExIx2 ... xk}for any X1 a) set E C H and xI, x2, ..., X, x E X. Abbreviate the leftsideof (8.1) by M(Exl,x2...xk)(x).Relativize Theorem 7.1 to a. This yields

M(A)(x) = M(A)(x)for every x E X. But

M*(A)(x) < M(A)(x), x e X,since a A1((a, x)) set is Borel. SoM*(A)(x) < M(A)(x)

for every x E X. The inequality in the reverse direction being obvious, wehave proved Theorem 1.1 when X = 2W.To complete the proof, suppose that X is a Borel (or even analytic)subsetofa Polish space. By the Borel isomorphism theorem, we may regard X withoutloss of generality as an analytic subset of 20 and also suppose that X : 21.Identify each element u of P(X) with the unique element of P(2W) whosetrace on X is ,u. It is now easy to see that P(X) is (can be identifiedwith) ananalyticsubset of P(2W) and hence that F is an analyticsubset of 2Wx P(2W).Fix x* E 2W- X and define a new gamblinghouse F on 2W as follows:

f(\J = I (X) U {5(X*)} if x E X,F {o x*)l if x E 2) - X.Since the set 17o= {(x, C5(x*))E 2W x P(2W): x E 2W} is Borel in 2W x P(20)and F = F u Fo, it follows that F is an analytic subset of 2W x P(2'). HenceF is an analytic gambling house on 2W. Let Al denote the optimal rewardoperator associated with F and let H = 20 x 20 x . .. . Let A be an analyticsubset of H = X x X x , x E X, and e > O. Plainly, A is analytic in H.So by the result establishedfor 2W in the previous paragraph, here is a Borel,open subset O' of H such that 0' contains A and

M(O')(x') < M(A)(x) + e,.Let 0 = 0' n H, so 0 is Borel and open in H and contains A. Since A C Hand x E X it is easy to see that M(A)(x) = M(A)(x). Hence,M(O)(x) < M(O')(x) < M(A)(x) + 8eThis completes the proof of Theorem 1.1. El

We conclude this section with the observation that Theorem 1.1 implies agenuine "capacitability"result for the set functions M*(-)(x), even thoughM*(*)(x) need not be a capacity [13].


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A CAPACITABILITY HEOREM N MEASURABLEGAMBLINGTHEORY 243Theorem 8.1. For every x E X and analytic subset A of H,

M* (A)(x) = sup{M* (K)(x): K C A &K is T1 compact}.Proof. Since M* is monotone, only the equation < needs to be established.This follows from Theorem 1.1 and the following string of inequalities:M(A)(x)= sup a(A)

aEY(x)= sup sup{fa(K): K C A &K is Ti-compact}aEX(x)= SUp SUp {a(K): K c A &K is Ti-compact}

aEX(x)= sup{M(K)(x): K C A &K is Ti-compact} inf{F(O)(x): O D A &O is open} > F(A)(x) .

Since the inequalityin the oppositedirectionis trivial,the theoremis proved. 0lThe rest of this section will be devoted to the extension of Theorem 9.1 tobounded, upper analytic functions on H (Theorem 1.2). The proof uses ideasfrom Monticino [14] (see, in particular,Theorem 4.2 in [14]).


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244 A. MAITRA,R. PURVES,AND W. SUDDERTHAs in Theorem 1.2, assume that g is a bounded, upper analytic function onH. For any strategy a, measurableor not, ag = f g da is well defined, sinceg is analytically measurable.For the same reason, if a is measurable, hen thevalue of ag, whether computed in the finitely additive or countably additive

mode, is the same.The first step in the proof of Theorem 1.2 is the uniform approximationof gby an upper analytic function with only finitely many values. Assume withoutloss of generalitythat 0 < g < 1 and fix e > 0. Let n(e) be the least integern such that ng > 1 and, for n = 1, 2,..., n(e), define An = {h E H: g(h) >ne}. Then setg =4eAI +A 2 + + An(o)]

where we are denoting the indicator function of the set Ai by the symbol Ai.The Ai are analytic. Thus g is upper analytic with values in {0, e, 2e, ...n(e)e} . Moreover, k < g < + e.Thus, it sufficesto prove Theorem 1.2 for 4 and, consequently, for e- 1.So, without loss of generality,assume from now on that(9.1) g =Al +A + ...+Awhere A1 D A2 D D An are analytic subsets of H.The next step is an approximation of each analytic set A by an analytic,G6 set G as was essentially done in ?6. Fix the analytic set A C H and let0 1-},where l(p) is the last coordinateof the finite sequence p and X* = UmEN xm.By virtue of Theorem 9.1, the set B would be the same if we replaced F by Min its definition. Consequently, B is an analytic subset of X* (see the proofof Lemma 6.3). Next define the analytic G3 set

G = G6(A)= {h E H: p,(h) E B infinitely often}.Lemma9.2. For all x E X, F(A n (H- G))(x) = 0.Proof. Suppose,by way of contradiction,that, for some x and a availableatx ,

a(A n(H-G)) >0.By the finitelyadditive Levy 0-1 law [20],

a[Pn(h)]((A n (H - G))pn(h)) -* 1for a-almost every h E A n (H - G). Hence, there is an h E H - G suchthat a[pn(h)](Apn(h))> 1 - e for infinitely many n. Since F(Apn(h))(hn)>?[Pn(h)](APn(h)),h E G, which yields the desired contradiction. 0

Now, for each Ai occurring n (9.1), define(9.2) Bi = B6(Ai), Gi = G6(Ai).Notice that B1 D B2 D ...D Bn and G1D G22 D Gn Now define g, onH and ue on XA by

ge = GI + G2+ * + Gne u. =B1 +B2 + .**+Bn


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A CAPACITABILITYHEOREM N MEASURABLEGAMBLINGTHEORY 245We note that both g, andu, are upper analytic functions. Finally, define, forx E X ,

(Tu,)(x) = suP{J U,(pt)da: a available at x, t a stop rule}.The notation Tu, is consistentwith that in [5] if we pass to the "partialhistoryhouse." In particular, the result of Strauch [22], as modified in [5, Lemma 4.2],implies that (Tu,)(x) would have the same value if the supremum were takenonly over measurable a availableat x and Borel stop rules t.The final steps of the argumentwill show that

Fg < Ig, < Tu, < Mg+(n+ 1)e.This will suffice to complete the proof of Theorem 1.2 since e is arbitrary, nis fixed, and, obviously, Mg < Fg.Lemma 9.3. Fg < Ig, .Proof. Let x E X and a be available at x. Then

ag = (A1) + (A2) +... + u(An)< a(GI) + a(G2)+ .+ a(Gn) = age,where the inequality holds because, for each i, a(Ai n (H - G1))= 0 by Lemma9.2. EnLemma 9.4. Ig, < Tu, .Proof. First notice that, for each h,g,(h) = GI(h) + G2(h) + . + Gn(h)

= lim Bl (Pk(h)) + lim B2 (Pk(h)) +.. + limBn Pk h))= lim(Bl + B2-+ .+ Bn (pk(h)).

To checkthe final equality, recallthat B1 D B2 D ... D Bn so that g, (h) = m ifand only if limk B1 pk(h)) = ... = limk Bm(Pk(h)) = 1 and limk Bm+I (Pk(h)) =* = liMkBn(pk(h))= 0 for m = 0, 1, .. .,n.Now, for any a available at x,

age = Jlim(Bi + B2 + + Bn)(Pk) da = limU,(Pk) da= lim u,(pt) da < sup J Ui(pt) da.

Here the final equality is a trivial variationof the Fatou equation [ 17, Theorem10.4] (see also [24] for a proof of a countably additive version of the Fatouequation;this proof can be adaptedto the finitelyadditive setting by using thefinitelyadditive Levy 0-1 Law [20]). Take the supreumumover a available atx to finish the proof. ElLemma 9.5. Tu, < Mg + (n + 1) .Proof. Fix x E X. Byvirtue of the remark mmediately followingthe definitionof Tue, we can choose a measurable a availableat x and a Borel measurablestop rule t such that


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246 A. MAITRA,R. PURVES,AND W. SUDDERTHNext we use a selection theorem [12, Lemma 2.1] to obtain analytically mea-surable mappings (7k: X* -+ P(H) such that Ak(P) E 1(l(p)) and, if p E Bk,then

Uk(p)(Akp) > 1- efor k =, 2, ..., n. Then define, for p E X*,

i(P) = afnP) if p E Bn,= ('n- 1(P) if pE Bn_ 1-Bn,

=J1(P) if peB1-B2,= v1 (p) (say) elsewhere.

Then a is analytically measurable from X* to P(H). Furthermore,sinceA1 D A2 D * D An, it follows that-(p)(Akp)> 1- if PEBk

for k = 1, 2,..., n. Consequently,-(Pt)(gPt) = i(pt)((Al + A2 + * + An)pt)> (BI + B2+ *+ Bn)(Pt) ne = u,(pt) - ne -

Now let a be the measurable strategy available at x which agreeswith aprior to time t and satisfies a[pt(h)] = -(pt(h)) for a-almost all h . ThenMg(x) > ag = Ja(pt)(gPt)da'

> Juc(pt)du - ne > Tu(x) - (n + 1)e. ElWe remark hat it is consistentwith ZFC that Theorem 9.1 does not extend tocoanalytic sets. Indeed, under the assumptionthat there exists a non-LebesguemeasurablePCA subset of [0, 1 , Monticino [ 14] constructed a Borel gamblinghouse F on a Polish space such that there is a coanalytic set C C H withF(C) :$ M(C). In other words, for some initial fortune x, nonmeasurablestrategiesprovide a distinctadvantageovermeasurable trategies. It also followsthat M*(C)(x) > M(C)(x) for some x E X.

10. APPROXIMATING FUNCTIONSIn this final section we prove an approximation result for bounded, upperanalytic functions on the set H of histories, which is analogousto the resultfor sets proved in Theorem 1.1.A subset of H which is a countable intersection of Borel, open subsets ofH will be called a special G6 set. The following theoremis the approximationresult for functions.

Theorem 10.1. Let F be an analytic gambling house on a Borel subset X of aPolish space. If g is a bounded, upperanalyticfunction on H, then(Mg)(x) = inf(Mf)(x),


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A CAPACITABILITYHEOREM N MEASURABLEGAMBLINGTHEORY 247where the infimumis overall functions f: H -+ 9i such that f > g, f takeson finitely many values, and {f > c} is a special G, for every c E 9i.Proof. It follows by virtue of the arguments of the previous section that itsuffices to prove the theorem when g is given by (9.1). Fix e > 0 and xo EX. Let the sets G, be defined by (9.2). Recall that each Gi is a countableintersection of analytic, open sets, and that, by virtue of Lemma 9.2,

M(Ai n (H-Gi))(x) =0 forallxe X.We now need a lemma.Lemma 10.2. There xist specialG, sets Gi, i = 1, 2, ... , n, withthe ollow-ing properties:

(a) GID G2D..D Gn,(b) Gi C di, i= 1, 2, ... , n, and(c) M(G1 + G2+ *+ G)(xo) < M(G1 + G2+ + Gn)(Xo)+e?

Proof. LetD= {(h, a) eHx [0, n]: G1(h)+G2(h)+ ..+Gn(h) >a}.

Plainly, D is analytic in H x [0, n]. So, By Corollary4.4 in [12], there is aBorel subset E of H x [0, n] such that D C E and(10.1) sup (cx A)(E)< sup (ox A)(D)+e,

uEl(xo) aEY(xo)where A is Lebesgue measure on [0, n]. Set 0(h) = A(Eh), h E H. Clearly,then, 0 is Borel measurable, 0 > G1 + G2 + * + Gn, and, because, of (10.1),(Mq)(xo) < M(GI + G2 + ... + Gn)(xo) + ?

Let Ci = {q > i}, i = 1, 2, ..., n, so Ci 2 Gi. Hence, by the proof ofCorollary 6.8 in [12], there is a special G, set Fi such that Ci D Fi D Gi . LetGi = F1 n F2 n ...n Fi, i = 1,2, .. ., n Then it is straightforward to verifythat the sets Gi satisfy conditions (a)-(c) of the lemma. 0

To continue with the proof of Theorem 10.1, let G1,G2, .. , Gnbe as inLemma 10.2. It follows that Ai n (H - Gi) C Ai n (H - Gi) so that

M(Ain (H- Gi))(xo)= 0.Since Ai n (H - Gi) is analytic, we can apply Theorem 1.1 to get a Borel, openset Oicontaining Ai n (H - Gi) such that

M(0')(xo) < l/n2, i = 1, 2, ...,n.Set Oi = OiUOil+ I U ...UOn, so 0i is Borel, open, 01202 D O?n and,by [13, Lemma 2.1],

M(0,)(xo) < M(Oi)(xo) + M(01,+)(xo) + * + M(0n)(xo) < e/ni =1,...,n.Define f =G1 +G2 + *+ Gn +01 + ?2 + + n -


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248 A. MAITRA,R. PURVES,AND W. SUDDERTHSince Al C Gi U 0i, it follows that f > g. Clearly, f takes on finitely manyvalues and {f > c} is a special G, for every c E 9l. Finally, for any a E I(xo),

af = (G + 2+ +Gn) + C( ) + (02) + ***+ ?(0)< a(Gi + G2 + ** + Gn) + M(Oi)(xo) + M(02)(xo) + ***+ M(On)(xo)< cU(dl+ d2 + ** + Gn)+6 .

Now take the sup over a E X(xo) to get(Mf)(xO)< M(GI+ G2+ + Gn)(XO)e

g and {f > c} is a Borel,openset for everyc e 9R.We concludewithan examplewhichshows hat this conjectures false.Example. Let X = {0, 1, 2, 3}, 17(0) = {(5(1)}, I(2) = {(5(2)}, 17(3) ={J(3)}, and 1(l) = {5(l), 236(2)+ 3a(3)}, where 5(x) denotes point massat x. Let Gi = {h E H: hn= i infinitelyoften}, i = 1, 2. Let g=G1+ 2(G2 G1). It is easyto seethat (Mg)(0) = 4. Nowlet f be a functionon H such that f > g and {f > c} is open for every c E 91. Then theset {f 1} is open and contains G1. Let T be a stopping ime suchthat{f > 1}={T 1,

6(1) if (xi = 0) or (xi = I &i h n),Ui(XI , x2, *- xi) = (xi) if xi = 2 or 3,

123a(2) + 3a(3) if xi = 1&i = n.It is noweasyto see that a({f > 1}) = 1 and a({f > 2}) > a(G2 G1)= 24Consequently,

(Mf)(0) > af > ({f > 1})+ ({f > 2}) > 3 *REFERENCES

1. D. P. Bertsekasand S. E. Shreve,Stochasticoptimalcontrol: The discretetime case, Aca-demic Press,New York, 1978.2. G. Choquet,Lectureson analysis,Vol. 1, Integrationand TopologicalVector Spaces,Ben-jamin, New York and Amsterdam,1969.


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A CAPACITABILITYHEOREM N MEASURABLEGAMBLINGTHEORY 2493. C. Dellacherie, Quelquesresultats ur les maisonsde jeux analytiques, Seminairede Prob-abilites XIX, Strasbourg1983-84), LectureNotes in Math., vol. 1123, Springer-Verlag,Berlin and New York, 1985, pp. 222-229.4. L. E. Dubins and L. J. Savage,Inequalities or stochasticprocesses,Dover, New York, 1976.5. L. E. Dubins, A. Maitra,R. Purves, and W. Sudderth,Measurable,nonleavablegamblingproblems,Israel J. Math. 67 (1989), 257-271.6. A. S. Kechris,Measure and category n effectivedescriptive et theory,Ann. Math. Logic5

(1973), 337-384.7. A. Louveau,Recursiveness nd compactness,Higher Set Theory(Proceedings,Oberwolfach,Germany,1977), LectureNotes in Math.,vol. 669, Springer-Verlag, erlin and New York,1978, pp. 303-337.8. -, Ensembles analytiqueset boreliens dans les espaces produits,Asterisque 78 (1980),1-84.9. __, Recursivity nd capacity heory,Proc. Sympos.Pure Math.,vol. 42, Amer. Math.Soc.,Providence,R.I., 1985, pp. 285-301.

10. A. Maitra,V. Pestien, and S. Ramakrishnan,Domination by Borel stopping imes andsomeseparationproperties,Fund. Math. 135 (1990), 189-201.11. A. Maitra,R. Purves, and W. Sudderth,Leavablegamblingproblemswith unboundedutili-ties, Trans.Amer.Math.Soc. 320 (1990), 543-567.12. __, A Borel measurableversion of K6nig'sLemma for randompaths, Ann. Probab. 19(1991), 423-451.13. -, Regularityof the optimalrewardoperator,University of Minnesota Schoolof Statis-tics, Tech.ReportNo. 534, 1989.14. M. Monticino, The adequacyof universal trategies in analyticgambling problems,Math.Oper.Res. 16 (1991), 21-41.15. Y. N. Moschovakis,Descriptive et theory,North-Holland,Amsterdam,1980.16. K. R. Parthasarathy,Probabilitymeasures on metric spaces,Academic Press, New York,1967.17. R. Purves and W. Sudderth,Some finitely additiveprobability,University of MinnesotaSchoolof Statistics, Tech. Report No. 220, 1973.18. , Somefinitelyadditiveprobability,Ann. Probab.4 (1976), 259-276.19. , How to stay in a set or K6nig's Lemmafor randompaths,Israel J. Math. 43 (1982),139-153.20. __, Finitely additivezero-one aws, SankhyaSer. A 45 (1983), 32-37.21. S. Saks, Theoryof the integral, Dover, New York, 1964.22. R. E. Strauch,Measurablegambling houses,Trans. Amer. Math. Soc. 126 (1967), 64-72(Correction130 (1968), 184).23. W. Sudderth, On the existenceof good stationarystrategies,Trans. Amer. Math. Soc. 135

(1969), 399-414.24. __, On measurablegamblingproblems,Ann. Math.Statist.42 (1971), 260-269.SCHOOL OF STATISTICS,UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MINNESOTA 55455DEPARTMENT OF STATISTICS, UNIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA 94720SCHOOL OF STATISTICS, UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MINNESOTA 55455

A Capacitability Theorem in Measurable Gambling Theory

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