Sure Wins, Separating Probabilities and the Representation of Linear Functionals
aa r X i v : . [ m a t h . F A ] O c t SURE WINS, SEPARATING PROBABILITIES AND THE REPRESENTATION OFLINEAR FUNCTIONALS
GIANLUCA CASSESE
Abstract.
We discuss conditions under which a convex cone
K ⊂ R Ω admits a finitely additive probability m such that sup k ∈K m ( k ) ≤
0. Based on these, we characterize those linear functionals that are representableas finitely additive expectations. A version of Riesz decomposition based on this property is obtained aswell as a characterisation of positive functionals on the space of integrable functions. Introduction
A long standing approach to probability, originating from the seminal work of de Finetti, views setfunctions P as maps which assign to each set (event) E in some class A the price P ( E ) for betting 1 dollaron the occurrence of E . A set function generating a betting system which admits no sure wins was termedcoherent by de Finetti who proved in [5] that a set function on a finite algebra A is coherent if and only if itis a probability. Since then this result has been extended and generalized by various authors, among whichHeath and Sudderth [9], Lane and Sudderth [10] and Regazzini [11], to name but a few; Borkar et al. [4] isa more recent example.In this paper we examine the absence of sure wins for a convex cone K of real valued functions on somearbitrary set Ω, obtaining conditions for the existence of a finitely additive probability measure m suchthat sup k ∈K m ( k ) ≤
0, i.e. a separating probability . The special case in which K is the kernel of somelinear functional leads to the characterization of those functionals that admit the representation as finitelyadditive expectations, a topic addressed by Berti and Rigo in a highly influential paper [2]. A version ofRiesz decomposition based on this representation property is obtained.Throughout the paper Ω will be a fixed set, 2 Ω its power set, R Ω and B the classes of real-valued andbounded functions on Ω respectively (the latter endowed with the topology induced by the supremum norm).All spaces of real-valued functions on Ω (e.g. bounded or integrable) will be considered as equipped withpointwise ordering, with no further mention. The lattice notation f + and f − wll be used to denote thepositive and negative parts of f ∈ R Ω . The term probability is used to designate positive, finitely additiveset functions m on 2 Ω (in symbols, m ∈ ba + ) such that m (Ω) = 1. The symbol P ba will be used to denotethe family of all probability measures; P the subfamily of all countably additive probability measures. If A ⊂ Ω then by S ( A ) and B ( A ) we denote the class of simple functions generated by A and its closurein B . We adopt the useful convention of identifying single-valued functions with their range so that, forexample, we may use 1 either to denote an element of R , or a function f on Ω such that f ( ω ) = 1 for all Date : October 28, 2018.2000
Mathematics Subject Classification.
Primary 28A25, Secondary 28A12, 28C05.
Key words and phrases.
Daniell theorem, Finitely additive probability, Finitely additive supermartingales, Integral repre-sentation of linear functionals, Riesz decomposition. ω ∈ Ω. In the terminology adopted throughout the following sections a sure win is defined to be an elementof R Ω which exceeds 1.We recall that f ∈ R Ω+ is integrable with respect to m ∈ ba + , in symbols f ∈ L ( m ), if and only if(1.1) sup { m ( h ) : h ∈ B , ≤ h ≤ f } < ∞ The integral m ( f ) coincides then with the left hand side of (1.1); moreover, f ∧ n converges to f in L ( m ) [7,theorem III.3.6]. A special notion of convergence in L ( m ) will be used in the following. A sequence h f n i n ∈ N issaid to converge orderly in L ( m ) to f if f n ∈ L ( m ) for all n and there exists a pointwise decreasing sequence (cid:10) ¯ f n (cid:11) n ∈ N in L ( m ) + which converges to 0 in L ( m ) and is such that | f n − f | ≤ ¯ f n for n ≥
1. It is easily seenthat if a sequence h f n i n ∈ N converges to f orderly in L ( m ) then so does any of each subsequences; moreover,the space of sequences converging orderly in L ( m ) is a vector space.2. Separating Probabilities
Fix a convex cone
K ⊂ R Ω (that is f + g, λf ∈ K whenever f, g ∈ K and λ ≥
0) and let K b = { k ∈ K : k − ∈ B } . For each f ∈ R Ω let U ( f ) = { α ∈ R : α + k ≥ f for some k ∈ K} and define π K : R Ω → R as(2.1) π K ( f ) = inf { α : α ∈ U ( f ) } From (2.1), π K is monotonic, π K ( λ + f ) = λ + π K ( f ) for each λ ∈ R and f ∈ R Ω and π K ( f ) ≤ sup ω ∈ Ω f ( ω )(as 0 ∈ K ). Since K is a convex cone, U ( f ) + U ( g ) ⊂ U ( f + g ) and U ( λf ) = λ U ( f ) for λ > π K is thussubadditive and positively homogeneous; moreover, π K ( k ) ≤ k ∈ K .Given that π K (0) = 2 π K (0) ≤ π K (1) = π K (0) + 1, then π K (0) > −∞ implies π K (0) = 0 and π K (1) = 1. Moreover there is k ∈ K such that k ≥ π K (1) ≤
0. Thus:
Lemma 1.
Let
K ⊂ R Ω be a convex cone. Then the following are equivalent: (i) π K (0) > −∞ , (ii) π K (0) = 0 , (iii) π K (1) = 1 , (iv) K contains no sure wins. Denote L ( π K ) = (cid:8) f ∈ R Ω : π K ( | f | ) < ∞ (cid:9) . It is clear that B ⊂ L ( π K ). Define also(2.2) M ( K ) = (cid:26) m ∈ P ba : K ⊂ L ( m ) , sup k ∈K m ( k ) ≤ (cid:27) and let M ( K b ) be defined likewise. We shall refer to elements of M ( K ) as separating probabilities for K . Itis clear that if m ∈ M ( K b ) then L ( π K ) ⊂ L ( m ). Proposition 1.
Let
K ⊂ R Ω be a convex cone. Then M ( K b ) is non empty if and only if K contains no surewins.Proof. Assume that K contains no sure wins. By Lemma 1 and the Hahn Banach Theorem, we may finda linear functional φ on B such that φ ≤ π K on B and φ (1) = 1. If f ∈ B + then φ ( f ) = − φ ( − f ) ≥− π K ( − f ) ≥
0. Therefore φ is positive and, since continuous [7, V.2.7], it may be represented as theexpectation with respect to some m ∈ P ba . If f ∈ L ( π K ) + , the left hand side of (1.1) is bounded by π K ( f )so that L ( π K ) ⊂ L ( m ). Then K b ⊂ L ( m ) and m ( k ) = lim n m ( k ∧ n ) ≤ π K ( k ) ≤ k ∈ K b so that m ∈ M ( K b ). If m ∈ M ( K b ) and k ∈ K is a sure win, then k ∈ K b and m ( k ) ≤
0, a contradiction. (cid:3) The functional π K is well known in mathematical finance under the name of superhedging price . URE WINS 3
A classical application of Proposition 1 considers the collection K of all finte sums of the form P n a n ( F n − λ ( F n )) where a , . . . , a N are real numbers, F , . . . , F N are elements of some A ⊂ Ω and λ : A → R . It isthen clear that K admits no sure wins if and only if there is m ∈ P ba such that m | A = λ . If the sums in K are allowed to admit countably many terms provided P n | a n λ ( F n ) | < ∞ , then m will possess the additionalproperty that m ( S n F n ) = P n m ( F n ) when h F n i n ∈ N is a disjoint sequence in A . This informal statementis essentially a reformulation of [9, theorems 5 and 6, p. 2074] . It admits an interesting generalisation tothe case of concave integrals, a special case of the monotone integral of Choquet treated, e.g., in [8]. Definition 1.
An extended real-valued functional γ on a convex cone L ⊂ R Ω is a concave integral if it ispositively homogeneous, monotone, superadditive and such that γ ( c + f ) = γ ( c ) + γ ( f ) when c, f ∈ L and c is a constant. If γ is a concave integral on L we define its core to be the set(2.3) Γ( γ ) = { λ ∈ ba + : L ⊂ L ( λ ) , γ ( f ) ≤ λ ( f ) , f ∈ L } The following Lemma is essentially a restatement of a result of Shapley [12, theorem 2, p. 18]. Itcharacterises the properties of a concave integral in terms of its core.
Lemma 2.
Let L ⊂ R Ω be a convex cone that contains the constants and is such that f ∈ L implies f + ∈ B . Let γ : L → R be a concave integral and γ (1) > . Then γ (1) < ∞ if and only if for each convexset C ⊂ L ∩ B such that γ ( C ) ≡ sup f ∈ C γ ( f ) < ∞ there exists λ C ∈ Γ( γ ) such that (2.4) sup f ∈ C λ C ( f ) = γ ( C ) Proof.
Assume, upon normalization, γ (1) = 1 and suppose that(2.5) α ( k − γ ( C )) ≥ N X n =1 ( f n − γ ( f n ))for some choice of α ≥ k ∈ C and f n ∈ L , n = 1 , . . . , N . The value under γ of the left hand side of (2.5)is less than 0 while that of the right hand side exceeds 1, contradicting monotonicity. Thus the collection K C of finite sums of the form P ≤ n ≤ N ( γ ( f n ) − f n ) + α ( k − γ ( C )) for α , k and f n , n = 1 , . . . , N as abovecontains no sure win; moreover, it is a convex cone of uniformly lower bounded functions on Ω. Accordingto Proposition 1, there exists λ C ∈ M ( K C ): thus, λ C ( f ) ≥ γ ( f ) for each f ∈ L (i.e. λ C ∈ Γ( γ )) and λ C ( k ) ≤ γ ( C ) whenever k ∈ C , proving (2.4). The converse is obvious. (cid:3) Lemma 2 has an interesting implication.
Corollary 1.
Let T be a collection of subsets of some set T , with { T } = τ ∈ T . For each τ ∈ T , let L τ be a linear subspace of B with ∈ L τ and φ τ a linear functional on L τ . The following are equivalent: ( i ) the collection ( φ τ : τ ∈ T ) is coherent in the sense that sup ( N X n =1 φ τ n ( b n ) : b n ∈ L τ n , N X n =1 b n τ n ≤ , N ∈ N ) < ∞ However we do not restrict A nor λ . Heath and Sudderth seem to suggest that the existence of m need not exclude surewins while it is clear that this cannot be the case. A less general version of this result was also proved, with different methods,in [4, theorem 2, p. 420] The inequality that follows is meant to hold pointwise in Ω × T GIANLUCA CASSESE ( ii ) there exists λ ∈ ba (Ω × T ) such that λ ( b τ ) = φ τ ( b ) for each b ∈ L τ and τ ∈ T Proof.
Assume ( i ) and define the functional γ on B (Ω × T ) implicitly as(2.6) γ ( b ) = sup ( N X n =1 φ τ n ( b n ) : b n ∈ L τ n , N X n =1 b n τ n ≤ b, N ∈ N ) It is readily seen that γ is monotone, superadditive and positively homogeneous. ( i ) implies that γ (1) < ∞ and that γ is real-valued while 1 ∈ L τ implies that γ is additive relative to the constants. ( ii ) follows from( i ), Lemma 2 and the fact that each L τ is a linear space: simply choose λ ∈ Γ( γ ). If λ is as in ( ii ) and P Nn =1 b n τ n ≤ b n ∈ L τ n n = 1 , . . . , N then P Nn =1 φ τ n ( b n ) = λ (cid:16)P Nn =1 b n τ n (cid:17) ≤ k λ k . (cid:3) Remark 1.
Writing τ ≤ υ when τ ⊂ υ makes of course T into a partially ordered set. If ( φ τ : τ ∈ T ) is coherent in the sense of Corollary 1 and if ( L τ : τ ∈ T ) is increasing in τ then necessarily φ υ | L τ ≥ φ τ whenever τ, υ ∈ T and τ ≤ υ . This conclusion has a direct application to the theory of finitely additivesupermartingales, treated in [6] . Much of this section rests on the conclusion, established in Proposition 1, that K b admits a separatingprobability in the absence of sure wins. This result, however, does not have an extension to K of a corre-sponding simplicity. To this end we shall need some results on the representation of linear functionals, to bedeveloped in the next section.3. The Representation of Linear Functionals
It is the purpose of this section to establish conditions for a linear functional φ on some linear subspace L of R Ω with 1 ∈ L to admit the representation(3.1) φ ( f ) = φ (1) m ( f ) f ∈ L for some m ∈ ba such that L ⊂ L ( m ), referred to as a representing measure for φ . We use the symbols K φ and K φb to denote the sets { f ∈ L : φ ( f ) = 0 } and { f ∈ K φ : f − ∈ B } , respectively. If φ (1) = 0, then K φb = { f − φ (1) − φ ( f ) : f ∈ L , f − ∈ B } . Thus if L is a vector sublattice of R Ω then m ∈ M ( K φb ) implies L ⊂ L ( m ) and φ ( f ) = φ (1) m ( f ) for every f ∈ L ∩ B (which clarifies the connection between separatingprobabilities and representing measures).The content of this section, as will soon become clear, owes much to the work of Berti and Rigo [2]. Theorem 1.
Let A ⊂ Ω be an algebra, µ ∈ ba ( A ) , L a vector sublattice of L ( µ ) with ∈ L and φ apositive linear functional on L . Denote by L ∗ the set of limit points of sequences from L which convergeorderly in L ( µ ) . The following are equivalent. ( i ) φ extends to a monotone function φ ∗ : L ∗ → R ; ( ii ) lim n φ ( h n ) = 0 whenever h h n i n ∈ N is a sequence in L which converges to orderly in L ( µ ) ; ( iii ) −∞ < lim n φ ( g n ) ≤ lim n φ ( f n ) < ∞ whenever h f n i n ∈ N and h g n i n ∈ N are sequences in L whichconverge orderly in L ( µ ) to f and g respectively, with f ≥ g ; ( iv ) φ admits a positive representing measure m such that m ∗ ( h ) ≡ lim n m ( h n ) exists in R and is uniquefor every sequence h h n i n ∈ N in L which converges to h orderly in L ( µ ) . URE WINS 5
Moreover, if φ is a positive linear functional on a vector sublattice L of R Ω with ∈ L then there exists aunique positive linear functional φ ⊥ on L such that φ ⊥ (1) = 0 and that (3.2) φ ( f ) = φ (1) m ( f ) + φ ⊥ ( f ) f ∈ L for some m ∈ ba + satisfying L ⊂ L ( m ) .Proof. Let h h n i n ∈ N be as in ( ii ) and (cid:10) ¯ h n (cid:11) n ∈ N be a decreasing sequence in L ( m ) converging to 0 in L ( m ) andsuch that ¯ h n ≥ | h n | , n = 1 , , . . . . Fix a sequence h α n i n ∈ N in R + such that lim n α n = ∞ . Any subsequence of h h n i n ∈ N admits a further subsequence (still denoted by h h n i n ∈ N for convenience) such that P n α n k h n k < ∞ .Fix η > h ηn = ( h n − η ) + , g ηk = X n ≤ k α n h ηn and g η = X n α n h ηn Then, (cid:8)P n>k α n h ηn > ǫ (cid:9) ⊂ { ¯ h k ≥ η } and (cid:13)(cid:13)(cid:13)P k
0, or m = 0 otherwise. Then,(3.4) φ ( f ) = lim n φ ( f ∧ n ) + lim n φ ( f − ( f ∧ n )) = φ (1) m ( f ) + φ ⊥ ( f )a conclusion which extends to general f ∈ L by considering f + and f − separately. The functional φ ⊥ ,as defined implicitly in (3.4), is clearly positive, linear and such that φ ⊥ (1) = 0. Decomposition (3.2) thusexists. If φ ( f ) = φ (1) v ( f ) + ψ ⊥ ( f ) were another decomposition such as (3.2), with v ∈ ba + , L ⊂ L ( v ) and ψ ⊥ a positive, linear functional on L with ψ ⊥ (1) = 0, then f ∈ L + would imply( φ ⊥ − ψ ⊥ )( f ) = lim n ( φ ⊥ − ψ ⊥ )( f − ( f ∧ n )) = φ (1) lim n ( m + v )( f − ( f ∧ n )) = 0which proves uniqueness of (3.2). Returning to the case L ⊂ L ( µ ), if ( iii ) holds, then it is obvious from (3.4)that φ ⊥ = 0; in addition the limit lim n m ( h n ) exists in R for each sequence h h n i n ∈ N in L which convergesorderly in L ( µ ) and does not depend but on the limit point h . (cid:3) One noteworthy implication of Theorem 1, obtained by replacing L with L ( µ ), is the following Theorem 2.
Let A ⊂ Ω be an algebra and µ ∈ ba ( A ) . Every positive linear functional φ on L ( µ ) admitsa positive representing measure m such that lim n m ( h n ) = 0 for every sequence h h n i n ∈ N in L ( µ ) whichconverges to orderly in L ( µ ) . Given that L ( µ ) is a normed Riesz space, its dual space is a vector lattice [1, theorem 12.1, p. 175]. ThusTheorem 2 also implies that continuous linear functionals, decomposing as the difference of two positivelinear functionals, admit a representing measure [2, theorem 7, p. 3255].Another application concerns more general functionals. In fact it is clear that the implication ( i ) → ( ii ) inTheorem 2 does not require φ to be linear. GIANLUCA CASSESE
Theorem 3.
Let L ⊂ R Ω be either (i) a Banach lattice containing the constants or (ii) L = L ( µ ) for some µ ∈ ba ( A ) and some algebra A ⊂ Ω . Assume that φ : L → R is a monotone functional such that (3.5) lim n inf { f ∈ L : φ ( f ) >η } φ ( nf ) = ∞ η > and, under (ii), (3.6) lim k ↓ sup f ∈ L { φ ( f ) − φ ( f − k ) } = 0 Then, lim sup n φ ( h n ) ≤ when h h n i n ∈ N converges to in norm or, under (ii), orderly in L ( µ ) . In particular,convex, monotone functionals on a Banach lattice are continuous.Proof. Each subsequence of h h n i n ∈ N contains a further subsequence for which it is possible to define g ηk and g η as in (3.3). Under ( i ), h g η i k ∈ N converges to g η in norm for all η ≥
0; under ( ii ) only for η >
0. In eithercase we conclude that φ ( g η ) ≥ φ ( α n h ηn ) ≥ φ ( α n ( h n − η )) and, given (3.5), lim inf n φ ( h n − η ) ≤
0. Choosing η = 0 under ( i ) or exploiting (3.6) under ( ii ) and recalling that the intial choice of the subsequence wasarbitrary, we conclude that lim sup n φ ( h n ) ≤
0. It is clear that a convex functional φ meets (3.5), (3.6) and,by monotonicity, | φ ( h ) − φ ( h n ) | ≤ φ ( | h n − h | ). (cid:3) Given the preceding results, it is now easy to extend Proposition 1 to K . Corollary 2.
Let
K ⊂ R Ω be a convex cone. Then M ( K ) is non empty if and only if there exist an algebra A ⊂ Ω and µ ∈ P ba ( A ) such that K ⊂ L ( µ ) and that the closure C µ of C = K − S ( A ) + in the normtopology of L ( µ ) admits no sure wins.Proof. If µ ∈ M ( K ) then µ is a separating measure for C µ which rules out sure wins. As for sufficiency,observe that ordinary separation theorems imply the existence of a continuous linear functional φ : L ( µ ) → R such that sup f ∈ C µ φ ( f ) ≤ φ (1). Given that K contains the origin, − S ( A ) + ⊂ C so that φ ispositive on S ( A ) and, since S ( A ) + is dense in L ( µ ) + and φ is L ( µ ) continuous, it is positive over thewhole of L ( µ ). The claim follows from Theorem 2. (cid:3) Corollary 2 is related to a result of Yan [13], where
K ⊂ L ( P ) and P is countably additive.The representation (3.1) extends beyond L ( µ ). Corollary 3.
Let L ⊂ R Ω be a linear space. A linear functional φ on L admits a representing measure ifand only if there exists µ ∈ ba such that L ⊂ L ( µ ) and φ is continuous with respect to the norm topologyof L ( µ ) . If, in addition, φ is positive and L a vector sublattice of R Ω , there exists a positive representingmeasure.Proof. The direct implcation is obvious. For the converse, let µ ∈ ba be as in the statement and denote by¯ φ the continuous, linear extension of φ to L ( µ ). If L is a vector lattice and φ is positive, the inequality φ ( f ) ≤ ¯ φ ( f + ) implies that such extension may be chosen to be positive and continuous. In either case theclaim follows from Theorem 2. (cid:3) Daniell theorem also follows easily.
Corollary 4.
Let L be a vector sublattice of R Ω containing and φ a positive linear functional on L .Then lim n φ ( f n ) = 0 for every sequence h f n i n ∈ N in L which decreases to pointwise if and only if φ admitsa representing measure m which is countably additive in restriction to the σ algebra generated by L . URE WINS 7
Proof.
Consider the case φ = 0, the claim being otherwise trivial. Then, by (3.2), φ (1) > φ admits arepresenting probability m . Let A = n E ⊂ Ω : inf { g ∈ L : g ≥ E } m ( g ) = sup { f ∈ L : f ≤ E } m ( f ) o and consider adecreasing sequence h E n i n ∈ N in A with T n E n = ∅ . For each η > h f n i n ∈ N and h g n i n ∈ N in L + with g n ≥ E n ≥ f n and m ( f n ) ≥ m ( g n ) − η − n . Let h n = inf { k ≤ n } f k . m ( h ) ≥ m ( g ) − η − ; if m ( h n − ) ≥ m ( g n − ) − η P n − k =1 − k for some n then, h n − + f n = h n + ( h n − ∨ f n ) ≤ h n + g n − implies m ( h n ) ≥ m ( f n ) + m ( h n − ) − m ( g n − ) ≥ m ( f n ) − η n − X k =1 − k ≥ m ( g n ) − η n X k =1 − k Thus the sequence h f n i n ∈ N may be chosen to be decreasing to 0 and such that m ( f n ) ≥ m ( g n ) − η for each n .Then, 0 = lim n m ( f n ) ≥ lim n m ( E n ) − η . It is well known that A is an algebra and that L ∩ B ⊂ B ( A ), seee.g. [3, p. 774]. Thus, m | A admits a countably additive extension to σ A and this, in turn, an extension µ to2 Ω . Since µ and m coincide on A , µ is another representing measure for φ . The converse is a straightforwardimplication of monotone convergence. (cid:3) References [1] C. D. Aliprantis, O. Burkinshaw (1985),
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