WHAT FINITE-ADDITIVITY CAN ADD TO DECISION THEORY

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1 Submitted to the Annals of Statistics WHAT FINITE-ADDITIVITY CAN ADD TO DECISION THEORY By Mark J. Schervish, Teddy Seidenfeld, Rafael Stern, and Joseph B. Kadane Carnegie Mellon University and Federal University of São Carlos We examine general decision problems with loss functions that are bounded below. We allow the loss function to assume the value. No other assumptions are made about the action space, the types of data available, the types of non-randomized decision rules allowed, or the parameter space. By allowing prior distributions and the randomizations in randomized rules to be finitely-additive, we prove very general complete class and minimax theorems. Specifically, under the sole assumption that the loss function is bounded below, we show that every decision problem has a minimal complete class and all admissible rules are Bayes rules. We also show that every decision problem has a minimax rule and a least-favorable distribution and that every minimax rule is Bayes with respect to the least-favorable distribution. Some special care is required to deal properly with infinite-valued risk functions and integrals taking infinite values. 1. Introduction. The following example, adapted from Example 3 of Schervish, Seidenfeld and Kadane (2009), is a case in which countablyadditive randomized rules do not contain a minimal complete class. It involves a discontinuous version of squared-error loss in which a penalty is added if the prediction and the event being predicted are on opposite sides of a critical cutoff. Example 1. A decision maker is going to offer predictions for an event B and its complement. The parameter space is Θ = {B, B C } while the action space is A = [0, 1] 2, pairs of probability predictions. The decision maker suffers the sum of two losses (one for each prediction) each of which equals the usual squared-error loss (square of the difference between indicator of event and corresponding prediction) plus a penalty of 0.5 if the prediction is on the opposite side of 1/2 from the indicator of the event. In symbols, AMS 2000 subject classifications: Primary 62C07; secondary 62C20 Keywords and phrases: admissible rule, Bayes rule, complete class, least-favorable distribution, minimax rule 1

2 2 SCHERVISH, SEIDENFELD, STERN AND KADANE the loss function equals L(θ, (a 1, a 2 )) = (I B a 1 ) 2 +(I B C a 2 ) { I[0,1/2] (a 1 ) + I (1/2,1] (a 2 ) if θ = B, I (1/2,1] (a 1 ) + I [0,1/2] (a 2 ) if θ = B C. To keep matters simple, we assume that no data are available, but one could rework the example with potential data at the cost of more complicated calculations. Figure 1 is a plot of all of the pairs (L(B, (a 1, a 2 )), L(B C, (a 1, a 2 ))), which shows all of the possible risk functions of non-randomized rules (pure strategies when there are no data available.) The admissible non-randomized Risks of Pure Strategies Risk at B complement Risk at B Fig 1. The risk functions of non-randomized rules in Example 1 strategies are the pairs (p, 1 p) for p [0, 1]. The corresponding points in Figure 1 are (i) from (0,3) up to but not including (0.5,1.5) in the upper left, corresponding to p [0, 1/2), (ii) (1,1) in the middle section, corresponding to p = 1/2, and (iii) from but not including (1.5,0.5) to (3,0) in the lower right, corresponding to p (1/2, 1]. The countably-additive randomized rules have risk functions in the convex hull of the set plotted in Figure 1. The resulting set is not closed, and its lower boundary is missing all points on the closed line segment from (0.5, 1.5) to (1.5, 0.5) except (1, 1). One consequence of these points being missing is

3 FINITELY-ADDITIVE DECISION THEORY 3 that there are many inadmissible rules, corresponding to points just above that line segment, that are dominated by other inadmissible rules, but not dominated by an admissible rule. In other words, the admissible rules do not form a complete class in this problem. If, however, one is willing to introduce finitely-additive randomizations, all of the missing risk functions of admissible rules become available. For example, there are finitely-additive probabilities P and P + on [0, 1] such that P (A) = 1 for every set of the form A = (1/2 ɛ, 1/2) and P + (C) = 1 for every set of the form C = (1/2, 1/2 + ɛ). Every missing point on the line segment between (0.5, 1.5) and (1.5, 0.5) is the risk function of a randomized rule that gives a 1 the distribution αp + (1 α)p + for some α [0, 1/2) (1/2, 1] and sets a 2 = 1 a 1. For example, the risk function for α = 0 is the point (0.5, 1.5) while the risk function for α = 1 is the point (1.5, 0.5). One result (Theorem 1) that we prove in this paper is that every decision problem with a loss function bounded below has a minimal complete class consisting of Bayes rules if finitely-additive randomizations are allowed. There are several complete class theorems in the countably-additive literature that make additional assumptions about the loss function (e.g., continuous, convex) and about the distributions of the available data (e.g., exponential family distributions). Some of these results can be found in Berger and Srinivasan (1978); Brown (1971) and Section 5.7 of Lehmann and Casella (1998). Another of our theorems is a general minimax theorem (Theorem 2) stating that every decision problem with a loss function bounded below has a minimax rule and least-favorable distribution such that all minimax rules are Bayes with respect to that least-favorable distribution. Heath and Sudderth (1972) prove a finitely-additive minimax theorem when risk functions are bounded. Heath and Sudderth (1978) consider cases in which a Bayesian analysis will be performed and the joint distribution of data and parameter can be computed by integration in both orders (a property not possessed by all finitely-additive distributions.) In these situations they present results about Bayes rules and extended admissible rules. Our results also cover cases in which the loss function is allowed to assume the value. An example of such a loss function is the logarithmic loss for predicting events. In the notation of Example 1, replace (I B a 1 ) 2 + (I B C a 2 ) 2 by log(a 1 [1 a 2 ])I B log(a 2 [1 a 1 ])I B C. Dealing with loss functions that assume the value requires special care. In particular, the set of functions that need to be integrated is not a linear space. The paper is organized as follows. We start with a general overview of

4 4 SCHERVISH, SEIDENFELD, STERN AND KADANE decision theory in Section 2. The next three sections give mathematical background needed to prove the main theorems. The background begins with topology and convergence in Section 3. At that point, we have enough tools to give an overview of the theory of integration with respect to finitelyadditive probabilities in Section 4. Because risk functions are integrals of loss functions with respect to various probability distributions, it is necessary to understand what it means to integrate a function with respect to a finitelyadditive probability. The final bit of background includes some results on separation of convex sets of unbounded functions in Section 5. After the mathematical background, we return to results specific to decision theory. Throughout the paper, we assume that the loss function is bounded below. No other assumption is made about the structural parts of the decision problem. Section 6 gives properties of the risk sets of general decision problems involving finitely-additive probabilities and randomizations with loss functions that are bounded below. The first of the main theorems is the complete class theorem (Theorem 1) that appears in Section 7. It states that every decision problem has a minimal complete class of rules consisting of admissible Bayes rules. The second theorem is Theorem 2 in Section 8, which says that every decision problem has a minimax rule and a least-favorable distribution. Also, each minimax rule is Bayes with respect to the least-favorable distribution. Section 9 gives some results on the existence and non-existence of Bayes rules with respect to specific types of priors. 2. General Decision Theory. Table 1 contains some notation that gets used frequently Decision Rules. Definition 1. A non-randomized rule is a function γ : X A. We denote by H 0 the set of all non-randomized rules that the decision maker has available. The risk function of a non-randomized rule δ is the function R(, γ) : Θ IR defined by (1) R(θ, γ) = L(θ, γ(x))p θ (dx). X A randomized rule δ is a coherent prevision on M H0. Its risk function is the function is (2) R(θ, δ) = R(θ, γ)δ(dγ). H 0 A randomized rule δ S M is called simple.

5 FINITELY-ADDITIVE DECISION THEORY 5 Table 1 Notation used throughout this document. The subscript Z on symbols like M Z, Λ Z, etc. will vary when referring to a specific set like Θ or H 0. Symbol Meaning A The action space X The space where available data take their values Θ The parameter space that indexes the set of data distributions P θ For each θ Θ, P θ is a probability measure on X IR IR { } endowed with the topology consisting of all open subsets of IR together with all sets of the form (c, ) { }. Z A generic set that serves as the domain of a class of IR-valued functions M Z The subset of IR Z consisting of those elements that are bounded below L(, ) The loss function mapping Θ A to IR and bounded below P Z The set of all coherent previsions (see Section 4) on M Z Λ Z The subset of P Z consisting of those coherent previsions that are minimum extensions (see Definition 12) of their restrictions to the bounded functions S Z The subset of Λ Z consisting of the minimum extensions (see Definition 12) of all simple probability measures on M Z, i.e., those supported on a finite set H 0 The subset of A X consisting of all non-randomized rules availble to the decision maker R(, δ) The risk function of a rule δ as defined in (2) R The set of risk functions for all finitely-additive randomized rules R 1 The subset of R whose randomizations are minimum extensions (see Definition 12) Each risk function is an element of M Θ. We can think of a non-randomized rule γ as the special case of a randomized rule δ with δ(a) = I A (γ), for A H 0. With this understanding, (2) is a generalization of (1). The operational understanding of a randomized rule δ is that a decision maker can select a γ from H 0 using an auxiliary randomization with distribution δ. We are deliberately vague about which elements of A X constitute the set H 0. The reason is that our results are general enough to cover whatever set H 0 one wishes to contemplate. One could choose all of A X, or just those rules that are formal Bayes rules with respect to various priors, or whatever subset of A X one wishes. The rules can arise from fixed-sample-size experiments or from sequential experiments. The only structure we assume for the decision problem is that the loss L is bounded below and that all finitely-additive randomizations (as defined in Definition 1) are allowed. Wald and Wolfowitz (1951) described two types of randomized rules. They refer to the randomizations we defined in Defintion 1, leading to risk functions of the form (2), as special randomizatons. They refer to a different kind of randomization, that is common in countably-additive decision the-

6 6 SCHERVISH, SEIDENFELD, STERN AND KADANE ory, as a general randomization. A general randomization is a mapping from X to probability distributions over A. In this manner, the randomization performed after observing a particular x can be virtually unrelated to the randomization that is performed after observing each other x. The theory proceeds by defining (3) L(θ, δ(x)) = L(θ, a)δ(x)(da), A and then inserting (3) directly into (1) to define the risk function of a general randomization δ as (4) R(θ, δ) = L(θ, a)δ(x)(da)p θ (dx). X A Special randomizations are special cases of general randomizations. For the case in which all of the P θ probabilities and all of the general randomizations are countably additive, we prove Lemma 29 (in Appendix A), that states that the risk functions that result from general randomizations are included in the set of risk functions that result from special randomizations. Aside from the restriction to countably additive probabilites, the major difference general and special randomizations involves the orders in which the integrals are performed. For general randomizations, the inner integral is over the randomization with the outer integral over the data distribution P θ. For special randomizations, the integral over the data distribution is inside, and the outer integral is over the randomization. For finitely-additive integrals, the order of integration matters to a larger extent than it does for countably-additve integrals Admissibility, Risk Sets, and Bayes Rules. Definition 2. Let Z be a set, and let f, g IR Z. We say that g dominates f if (i) g(z) f(z) for all z Z and (ii) there is z such that g(z) < f(z). The lower boundary of a subset A of IR Z, denoted L A, is the set of all functions f A such that there is no g A that dominates f. The lower boundary of a set A of functions is defined in terms of the closure of A. If more than one topology is available, one needs to be clear about which closure one is using. The topology of pointwise convergence of functions is always available for a function space since it is the product topology obtained by identifying each function from a space S 1 to S 2 as an element of S S 1 2. If the functions are all bounded, the topology of uniform convergence is also available. Since our results will apply to general function spaces, we will use only the topology of pointwise convergence.

7 FINITELY-ADDITIVE DECISION THEORY 7 Definition 3. The risk set R is the set of all risk functions of decision rules. A decision rule δ dominates another decision rule δ if R(, δ) dominates R(, δ ). A rule δ is admissible if no other rule dominates δ. A subset C of the set of all decision rules is called a complete class if, for every δ C, there exists a δ C such that δ dominates δ. A set C is an essentially complete class if, for every δ C, there exisits δ C such that R(θ, δ) R(θ, δ ) for all θ. A(n essentially) complete class is minimal if no proper subset is (essentially) complete. For the remainder of this paper, the risk set R is the set of all risk functions of randomized rules as defined in Definition 1. In addition to (as well as related to) the risk function, the Bayes risk of a decision rule with respect to one or several prior distributions is important to decision making. A prior distribution λ over Θ can also be interpreted (as in Section 4) as a coherent prevision on M Θ, hence as an element of P Θ. Definition 4. If δ is a randomized rule, the Bayes risk of δ with respect to λ is the value (5) r(λ, δ) = R(θ, δ)λ(dθ). Let r 0 (λ) = inf δ r(λ, δ). If (6) r(λ, δ 0 ) = r 0 (λ), then δ 0 is called a Bayes rule with respect to λ. Θ 3. Topology Background. Our results rely on the theory of product sets, nets, compact sets, and ultrafilters Product Sets. Definition 5. Let {Z α } α ℵ be a collection of sets, each of which has a topology. The product topology on the product set α ℵ Z α is the topology that has as a sub-base the collection of all sets of the form α ℵ Y α where Y α = Z α for all but at most one α, and for that one α, Y α is an open subset of Z α. If we let Z = α ℵ Z α, then the product set α ℵ Z α can be thought of as the set of all functions f : ℵ Z such that f(α) Z α for all α. If all Z α are the same set Z, then the product set is often written Z ℵ.

8 8 SCHERVISH, SEIDENFELD, STERN AND KADANE Definition 6. Let {Z α } α ℵ be a collection of sets. For each β ℵ, the function f β : α ℵ Z α Z β defined by f β (g) = g(β) is called an evaluation functional or a coordinate-projection function. It is not difficult to show that the product topology has two other equivalent characterizations: (i) the smallest topology such that all of the evaluation functionals are continuous, and (ii) the topology of pointwise convergence of the functions in the product set. Every open set in a topological space is the union of arbitrarily many basic open sets. Each basic open set is the intersection of finitely many sub-basic open sets. Hence, the following result is straightforward. Proposition 1. For every open set N in a product space, there exist a finite integer n, points α 1,..., α n ℵ, and open sets N j Z αj for j = 1,..., n such that N contains (7) {f : f(α j ) N j, for j = 1,..., n}. In particular, every neighborhood of a function g in the product space must contain a set of the form (7) such that g(α j ) N j for j = 1,..., n Nets. Our main use for nets is to identify the closure of a set. Definition 7. A partial order on a set D is a binary relation D with the following properties: (i) for all η D, η D η (reflexive), and (ii) if η D β and β D γ, then η D γ (transitive). A directed set is a set D with a partial order D that has the following additional property: (iii) for all η, β D, there exists γ D such that η D γ and β D γ. Definition 8. Let D be a directed set with partial order D. A function r : D T is called a net on D in T. Such a net is denoted either (D, r) or x = {x η } η D, where x η = r(η) for η D. If {x η } η D is a net, then for each η D, the set A η = {x β : η D β} is called a tail of x. If T is a topological space, we say that the net x converges to x T if every neighborhood of x contains a tail of x. Note that the set ZZ of positive integers is a directed set, and each sequence is a net on ZZ. Every result that holds for all nets holds a fortiori for all sequences. There are nets that are not sequences.

9 FINITELY-ADDITIVE DECISION THEORY 9 Example 2. Let D be the collection of all finite subsets of an uncountable set X. Create the partial order D on D defined by η 1 D η 2 if η 1 η 2. Notice that η j D (η 1 η 2 ) for j = 1, 2, making D a directed set. Let T be the collection of all indicator functions of subsets of X, and let x η = I η for each η D. It is straightforward to show that {x η } η D converges to I X. It is also easy to see that no sequence of indicators of finite sets can converge to the indicator of an uncountable set. The following results are taken from Dunford and Schwartz (1957). Proposition 2 (I.7.2, p. 27). limits of nets in C that converge. The closure of a set C is the set of all Proposition 3 (I.8.2, p. 32). The product of a collection of Hausdorff spaces is Hausdorff in the product topology. Definition 9. Let x = {x η } η D be a net in a topological space T. We call a net y = {y γ } γ D a subnet of x if there exists a function h : D D with the following properties: (i) y γ = x h(γ) for all γ D, (ii) γ 1 D γ 2 implies h(γ 1 ) D h(γ 2 ) and (iii) for every η D there exists γ D such that h(γ) D η. A cluster point of x is a point p T such that, for every neighborhood N of p and every η D, there exists η D such that η D η and x η N. It is trivial to see that a net is a subnet of itself. If we think of a sequence as a net, then it is also trivial that a subsequence is a subnet. Lemma 1. For a convergent net in a Hausdorff space, there is a unique limit that is also the unique cluster point. Proof. Clearly, the limit of a convergent net is a cluster point. The proof will be complete if we prove that every cluster point equals every limit. Assume to the contrary that there exists a cluster point p that is not equal to a limit p of a convergent net {x η } η D. Let N and N be disjoint neighborhoods of p and p respectively. There exists η 0 D such that x η N for all η D η 0. Because p is a cluster point, there exist η D such that η 0 D η and x η N. But x η N also, which contradicts N N =. Lemma 2. Let x = {x η } η D be a net in a topological space T, and suppose that p is a cluster point. Then there exists a subnet {y γ } γ D of x that converges to p.

10 10 SCHERVISH, SEIDENFELD, STERN AND KADANE Proof. For each neighborhood N of p and each η D, let α N,η D be such that η D α N,η and x αn,η N. Let D = {(N, η) : η D and N is a neighborhood of p}. We show next that D is a directed set. Define D by (N 1, η 1 ) D (N 2, η 2 ) to mean N 2 N 1 and η 1 D η 2, which is clearly transitive and reflexive. If (N 1, η 1 ), (N 2, η 2 ) D, let η D be such that η j D η for j = 1, 2. Then for j = 1, 2 (N j, η j ) D (N 1 N2, η ) D. Let h : D D be h(n, η) = α N,η, which clearly satisfies properties (ii) and (iii) in the definition of subnet. Let y (N,η) = x αn,η for each (N, η) D. To see that {y N,η } (N,η) D converges to p, let N 0 be a neighborhood of p. Then y (N,η) N 0 for all N N 0 and all η D. Lemma 3. same limit. A net converges if and only if every subnet converges to the Proof. The if direction is trivial because the net itself is a subnet. For the only if direction, assume that the net {x η } η D converges to p. Let {y γ } γ D be a subnet with corresponding function h. Let N be a neighborhood of p. Let η N be such that for all η D η N, x η N. Let γ N D be such that h(γ N ) D η N. If γ D γ N, then h(γ) D h(γ n ) η N and y γ = x h(γ) N. The following results are straightforward. Proposition 4. Let x = {x η } η D be a net in a topological space T with a subnet {y γ } γ D. Let f : T V be a function to another topological space V. Then {f(x η )} η D is a net in T with a subnet {f(y γ )} γ D. Proposition 5. Let Z be a set, and let {f η } η D and {g η } η D be convergent nets in IR Z. Let f and g be the respective limits. If f η (z) g η (z) for all z Z and η D, then f(z) g(z) for all z Z Compact Sets. The following results are taken from Dunford and Schwartz (1957). Proposition 6 (I.7.9, p. 29). if every net has a cluster point. A topological space is compact if and only Proposition 7 (I.5.7(a), p. 17). compact. A closed subset of a compact space is

11 FINITELY-ADDITIVE DECISION THEORY 11 Proposition 8 (I.8.5, p. 32). The product of a collection of compact spaces is compact in the product topology. It is well known that a subset of IR is compact if and only if it is closed and bounded. Lemma 4. subset of IR. For each real number b, set [b, ] is a compact Hausdorff Proof. First, we prove that IR is Hausdorff, hence so is every subset. Let x y IR. If both are finite then the following open intervals (x ɛ, x + ɛ) and (y ɛ, y + ɛ) are disjoint, where ɛ < x y /2. If x < = y, then (x 1, x + 1) and (x + 2, ] are disjoint open intervals. Next, we prove that [a, ] is compact. Let A = {A α } α ℵ be an open cover of [b, ]. Since IR, there exists α 0 ℵ such that A α0 contains a set of the form (c, ]. For the other elements of A, let B α = A α \ { }. Then {B α } α α0 is an open cover of [b, c], which has a finite subcover, because [b, c] is a compact subset of IR. That finite subcover, together with A α0 forms a finite subcover of [b, ] from A. The supremum and infimum are functionals defined on spaces of IR-valued functions. They are not continuous in general, but they do have the following property. Lemma 5. some set Z. Let {f η } η D be a convergent net (with limit f 0 ) in IR Z for If sup z Z f η (z) c η for all η and {c η } η D converges to c 0, then sup z Z f 0 (z) c 0. If inf z Z f η (z) c η for all η and {c η } η D converges to c 0, then inf z Z f 0 (z) c 0. Proof. Consider the claim about the supremum first. The claim is vaccuous if c 0 =. If c 0 <, suppose, to the contrary, that sup z Z f 0 (z) > c 0. Then there exists ɛ > 0 and z 0 Z such that f 0 (z 0 ) > c 0 + ɛ. Because f η converges to f 0, there exists η 0 such that η D η 0 implies that f η (z 0 ) > c 0 +2ɛ/3. Since c η converges to c 0, there exists η 1 such that η D η 1 implies that c η < c 0 +ɛ/3, hence f η (z 0 ) < c 0 +ɛ/3. There exists η 2 such that η j D η 2 for j = 0, 1. Hence, η D η 2 implies both f η (z 0 ) > c 0 + 2ɛ/3 and f η (z 0 ) < c 0 + ɛ/3, a contradiction. A similar argument works for the claim about the infimum.

12 12 SCHERVISH, SEIDENFELD, STERN AND KADANE Lemma 5 has a useful corollary. Corollary 1. Let c IR. Then {f : inf z Z f(z) c} and {f : sup z Z f(z) c} are closed Ultrafilters. Definition 10. Let S be a set and let U be a non-empty collection of subsets of S. We call U an ultrafilter on S if (i) A U and A B implies B U, (ii) A, B U implies A B U, and (iii) for every A S, A U if and only if A C U. An ultrafilter on S is principal if there exists s S such that U consists of all subset of S that contain s. Such an s is called the atom of U. Other ultrafilters are called non-principal. The following result is from Comfort and Negrepontis (1974), and it gives general conditions under which ultrafilters exist. Proposition 9 (Theorem 2.18, p. 39). Let B be a Boolean algebra, and let F B. If F has the finite intersection property, then there is an ultrafilter on B that contains F. Ultrafilters on directed sets are the main ones that we will need. If D is a directed set, then the collection of all tails of D has the finite-intersection property. (Every finite subcollection has non-empty intersection.) By Proposition 9, there is an ultrafilter U D that contains all tails of D. 4. Coherent Previsions Unbounded and Infinite-Valued Functions. De Finetti (1974) laid out the theory of coherent previsions for bounded random variables in great detail. In particular, a finitely-additive probability can be defined on an arbitrary collection of subsets of a general set Z, including the power set. For this reason, measurability conditions are often not included in theorems about finitely-additive probabilities. In this paper, we need an extension of the finitely-addtivie theory from bounded random variables both to unbounded random variables, as was done by Schervish, Seidenfeld and Kadane (2014), and to random variables that assume the value as well. We stop short of defining coherent previsions for random variables that both assume the value and are unbounded below. Definition 11. Let Z be a set, and let F be a subset of IR Z. Let P : F IR { }. An acceptable gamble is a linear combination of the

13 form (8) G(z) = FINITELY-ADDITIVE DECISION THEORY 13 n α j [X j (z) c j ], j=1 where each X j F, c j = P (X j ) for each j such that P (X j ) is finite, α j 0 for each j such that either P (X j ) = or there is z such that X j (z) =, and α j 0 for each j such that P (X j ) =. We call P a coherent prevision on F if sup z G(z) 0 for every acceptable gamble G. The conditions involving infinite values are needed to prevent from appearing in the formula for an acceptable gamble. If P is a coherent prevision on F, X F, and X IR Z, then P ( X) = P (X) is coherent with P even if X F and even if P (X) {, }. (Note that such an X could not assume the value.) Indeed, every acceptable gamble that involves P (X j ) = is equal to one with X j replaced by X j, so we could define coherence without allowing previsions in the formula for an acceptable gamble. For a set F containing functions that are all bounded below, a coherent prevision cannot take the value. The following result allows us to extend coherent previsions to sets that are either convex or linear. Proposition 10. Let P be a coherent prevision on a subset F IR Z consisting of functions that are bounded below. Then it is coherent to extend P to the convex hull of F by P (α 1 X α n X n ) = n j=1 α jp (X j ) for all finite n, all X 1,..., X n F and all non-negative α 1,..., α n that add to 1. It is also coherent to extend P to all linear combinations of the form α 1 X α n X n where each X j F satisfies P (X j ) < and α j 0 for each j for which there exists z such that X j (z) =. The extension is P (α 1 X α n X n ) = n j=1 α jp (X j ). For bounded functions, we have the following characterization of coherent previsions. Proposition 11. Let F be a linear space of bounded functions. Then P is a coherent prevision on F if and only if P is a linear functional that satisfies (i) P (1) = 1, and (ii) P (X) 0 for all non-negative X. The following trivial result is useful, but it illustrates the limitations associated with previsions of unbounded functions.

14 14 SCHERVISH, SEIDENFELD, STERN AND KADANE Lemma 6. If F is a set of unbounded functions that are all bounded below, there is a coherent prevision P with P (f) = for all f F. Proof. In (8), every α j 0 is required, hence (8) is unbounded above so long as at least one α j 0. The following result says that a coherent prevision on the set of all bounded random variables can be extended, in one operation, to all random variables bounded below. This is in contrast to the fundamental theorem of prevision, which allows more general extensions, but those more general extensions require that previsions be assigned to new random variables one at a time. Heath and Sudderth (1978, eqn. 1.2) state a similar result without being explicit about the possibility of the extension taking infinite values. Lemma 7. Let J be a linear space of bounded real-valued functions defined on a set Z, and let P be a coherent prevision on J. Let K J be a subset of M Z such that, for each X K and each real m 0, X m J. For each X K, define P (X) = sup P (Y ). Y J, Y X Then P = P on J. Let K 0 be the set of all linear combinations of elements of K J of the form n j=1 α jx j where α j > 0 for each j such that P (X j ) = or such that {z : X j (z) = } =. Define Q(Y ) = n j=1 α jp (X j ) for Y K 0. Then Q is well-defined and is a coherent prevision on K 0. Proof. Because P is coherent, Y X implies P (Y ) P (X) for X, Y J. Hence P (X) = P (X) for X J. Also, P (αx) = αp (X) if α > 0. For this reason, there is no loss of generality in assuming that all α j are 1 or 1. Next, we show that P (X 1 + X 2 ) = P (X 1 ) + P (X 2 ) for X 1, X 2 K. Notice that P (X) = lim m P (X m) because every bounded Y X is bounded above by X m where m = sup z Y (z). Then notice that, for all m, (X 1 m/2) + (X 2 m/2) (X 1 + X 2 ) m (X 1 m) + (X 2 m). The limits of the left-hand and right-hand expressions are both P (X 1 ) + P (X 2 ) while the limit of the middle expression is P (X 1 + X 2 ). Next, we show that Q is well-defined. We need to show that, (i) if X = n j=1 α jx j with α j = 1 for each j such that P (X j ) = or {z : X j (z) = }, (ii) Y = m k=1 β ky k (z) with β k = 1 for each k such that P (Y k ) =

15 FINITELY-ADDITIVE DECISION THEORY 15 or {z : Y k (z) = }, and (iii) X(z) = Y (z) for all z, then Q(X) = Q(Y ). Rewrite the equation n m α j X j = β k Y k, j=1 by moving all terms with 1 coefficients to the other side. The result has the form Z Z l = W W s, where each Z i is either an X j with α j = 1 or a Y k with β k = 1, and each W i is either a Y k with β k = 1 or an X j with α j = 1. Also, all of the Z i and W i are in K. We just proved that P is additive on K, so k=1 P (Z 1 ) + + P (Z l ) = P (W 1 ) + + P (W s ). Now, move the terms that corresponded to negative coefficients (all of which are finite) back to their original sides of the equation, and we get that Q(X) = Q(Y ). Finally, we prove that Q is coherent. Let K 1 be that part of K 0 where Q is finite and each element of K 1 is real-valued. From what we ve already shown, it follows that, on K 1 (i) Q is a linear functional, (ii) Q(1) = 1, and (iii) Q(X) 0 if X 0. It follows that Q is a coherent prevision on K 1. The only case of gambles not yet covered is that involving both elements of K 1 and functions with infinite prevision and/or functions that take the value. Let Q(X 1 ) =, let Q(X 2 ) be finite, and let X 3 take the value with Q(X 3 ) finite. We need to show that sup z [X 1 (z) c + X 2 (z) Q(X 2 ) + X 3 (z) Q(X 3 )] 0 and sup z [X 1 (z) c + X 2 (z) Q(X 2 )] 0. First, note that sup[x 1 (z) c + X 2 (z) Q(X 2 ) + X 3 (z) Q(X 3 )] = > 0. z Finally, let Y J be such that Y X 1 and > P (Y ) > c. Then, sup[x 1 (z) c + X 2 (z) Q(X 2 )] sup[y (z) P (Y ) + X 2 (z) Q(X 2 )] z z = sup[y (z) Q(Y ) + X 2 (z) Q(X 2 )], z which is non-negative because Y and X 2 are in K 1. Definition 12. We refer to Q in Lemma 7 as the minimum extension of P. If a prevision Q is the minimum extension of its restriction to the bounded random variables, then we say that Q is a minimum extension.

16 16 SCHERVISH, SEIDENFELD, STERN AND KADANE The term minimum extension is used because it assigns the minimum of all possible coherent previsions that are consistent with P to all functions that are bounded below. It is straightforward to see that the measuretheoretic definition of expectation with respect to a countably-additive probability is a minimum extension. Lemma 8. Let X be bounded below. Let P be a minimum extension such that P (X) <. Then P (X) = lim m P [XI {X m} ]. Proof. Since P [XI {X m} ] is non-decreasing in m, it converges to some number c 1 P (X). Hence mp (X > m) converges to c 2 = P (X) c 1 0. We need to prove that c 2 = 0. Assume, to the contrary, that c 2 > 0. Then Also, lim n (9) P (X) 2 n P (X > 2 n ) 2 n 1 P (X > 2 n 1 ) = 1. 2 n 1 P (2 n 1 < X 2 n ). n=1 For each n 1, let d n = 2P (X > 2 n )/P (X > 2 n 1 ). We ve assumed that lim n d n = 1. So P (2 n 1 < X 2 n ) = P (X > 2 n )(1 + ɛ n ) where lim n ɛ n = 0. It follows from (9) that P (X) 2 n 1 P (X > 2 n 1 )(1 + ɛ n ) =, n=1 which contradicts P (X) <. Lemma 9. Let X be bounded below. Let P be a minimum extension such that P (X) <. Let A = {X = }. Then P (XI A ) = 0 and P (A) = 0. Proof. Lemma 8 tells us that P (XI A ) = lim m P (XI A I {X m} ), but I A I {X m} = 0, so P (XI A I {X m} ) = 0 for all m and P (XI A ) = 0. If P (A) > 0, then P ([X m]i {X>0} ) P ([X m]i A ) mp (A), which goes to infinity as m, contradicting P (X) <. Since we make use of Lemma 6 in the proof of Theorem 2, we should note that the coherent prevision in Lemma 6 will not in general be a minimum extension.

17 FINITELY-ADDITIVE DECISION THEORY 17 Example 3. Let Z be an uncountable set. Then Z contains uncountably many disjoint countable subsets. Let B be an uncountable collection of countable disjoint subsets of Z. For each B B, define the unbounded function f B (z) = j=1 ni B(z n ), where B = {z 1, z 2,...} of Z. The minimum extension of a probability λ on Z assigns positive prevision to f B if and only if λ(b) > 0. At most countably many elements of B can have positive probability, hence at most countably many of the functions f B can have positive (let alone infinite) prevision under the minimum extension of λ Ultrafilter Probabilities. There is a correspondence between ultrafilters and 0-1-valued probabilities. Lemma 10. Let Z be a set, and let F be a field of subsets of Z. A finitely-additive probability P defined on F takes only the values 0 and 1 if and only if (i) P can be extended to P defined on 2 Z and (ii) there is an ultrafilter U of subsets of Z such that P (E) = 1 if and only if E U. Proof. For the if direction, the restriction of P to F takes only the values 0 and 1. For the only if direction, V = {E F : P (E) = 1} has the finite-intersection property, hence there is an ultrafilter U of subsets of Z such V U. Define P (E) = 1 if E U and P (E) = 0 if E U. It is clear that P is finitely-additive on 2 Z. Definition 13. Let U be an ultrafilter of subsets of some set Z. We call the probability P defined by P (E) = 1 if E U and P (E) = 0 if E U the probability corresponding to U. Lemma 11. Let D be a set. Let U be an ultrafilter of subsets of D. Let P be the minimum extension of the probability on D that corresponds to U. Then, for each f M D, f(η)p (dη) = sup inf f(η) = inf sup f(η). B U η B B U η B D Proof. Let k 0 = D f(η)p (dη). Since P (B) = 1 for all B U, Lemma 9 implies that k 0 sup η B f(η), for all B U. Similarly, k 0 inf η B f(η), for all B U. It follows that (10) sup inf f(η) k 0 inf B U η B sup B U η B f(η). If the two endpoints, call them a b, of (10) are not equal, then for each c (a, b) precisely one of {η : f(η) c} or {η : f(η) > c} is in U. If it is the

18 18 SCHERVISH, SEIDENFELD, STERN AND KADANE first of these, it contradicts b = inf B U sup η B f(η). If it the second one, it contradicts a = sup B U inf η B f(η). Hence a = b, and both inequalities in (10) are equality. An alternative expression for D f(η)p (dη) is f(η)p (dη) = sup{x : f 1 ([x, ]) U} = inf{x : f 1 ((, x]) U}. D The proof is similar to the proof of Lemma 11. Lemma 12. Let Z be a set. Let D be a directed set and let f = {f η } η D be a net in M Z. An element g of IR Z is a cluster point of f if and only if there is an ultrafilter U that contains all tails of D whose corresponding probability on D has minimum extension P such that g(z) = D f η(z)p (dη) for all z Z. Proof. For the only if direction, assume first that f converges to g. Let U be any ultrafilter that contains all tails of D, and let P be the minimum extension of the corresponding probability on D. Let h(z) = D f η(z)p (dη) for each z Z. We need to show that, for each z Z and each neighborhood N of g(z), h(z) N. Let z Z. Lemma 11, applied to ultrafilters of subsets of D, says that (11) sup{η : f η (z) B U inf η B f η(z) = h(z) = inf sup B U η B f η (z). If g(z) is finite, let ɛ > 0, and let N be the interval (g(z) ɛ, g(z) + ɛ). Because f converges to g, there exists η D such that f β (z) N for all β D η. Let B = A η = {β D : η D β}, which is in U. It follows that the left and right sides of (11) are respectively at least g(z) ɛ and at most g(z) + ɛ. Hence h(z) N. If g(z) =, let N = (c, ]. Then there exists η D such that f β (z) > c for all β D η. Let B = A η, which is in U. It follows that the left side of (11) is at least c, so that h(z) N. Next, assume that g is merely a cluster point of f. Then there exists a subnet f = {r τ } τ D that converges to g. The previous argument shows that for every ultrafilter U on D that contains all tails of D, g(z) = D r τ (z)p (dτ), where P is the probability that corresponds to U. Let h : D D be the cofinal function that embeds D in D, preserving order. Then r τ = f h(τ), so that g(z) = D f h(τ) (z)p (dτ). Let P be the probability on D induced by h from P. Then f η (z)p (dη) = f h(τ) (z)p (dτ). D D

19 FINITELY-ADDITIVE DECISION THEORY 19 For the if direction, let U be an ultrafilter that contains all tails of D such that g(z) = D f η(z)p (dη) for all z Z, where P is the minimum extension of the probability on D that corresponds to U. We need to show that g is a cluster point of f. Specifically, we need to show that, for every neighborhood N of g and η D, there exists β D η such that f β N. It suffices to prove this claim for all neighborhoods of the form N = {h : h(z j ) N j, for j = 1,..., n}, for arbitrary positive integer n, distinct z 1,..., z n Z and neighborhoods N 1,..., N n of the form N j = (g(z j ) ɛ, g(z j ) + ɛ) for ɛ > 0 if g(z j ) is finite and N j = (c, ] if g(z j ) =. So, let N be of the form just described, and let η D. We need to find β D such that η D β and f β (z j ) N j for j = 1,..., n. We know that, for all z Z, sup B U inf f β(z) = g(z) = inf β B sup B U β B f β (z). For each j such that g(z j ) is finite, let B j U be such that such that inf f β (z j ) > g(z j ) ɛ, η B j sup f β (z j ) < g(z j ) + ɛ. η B j For each j such that g(z j ) =, let B j U be such that such that inf f β (z j ) > c. β B j If we replace B j by B j A η, all of the last three inequalities above continue to hold. For each j, let β j B j, and let β D β j for all j. Then η D β and f β (z j ) N j for all j. Corollary 2. Let Z be a set. Let D be a directed set and let f = {f η } η D be a net in IR Z. The net f converges to g if and only if for every ultrafilter U that contains all tails of D with corresponding probability on D having minimum extension P, g(z) = D f η(z)p (dη) for all z Z Topology of Coherent Previsions. The set of coherent previsions is a set of functions defined on the space of random variables (functions) under consideration. In general, if Z is a space and we think of M Z as the set of all IR-valued functions that are bounded below, we can think of P Z as a set of functions from M Z to IR. As such, P Z has a product topology, which is also the topology of pointwise convergence. That is, a net {Q η } η D in P Z converges to R if and only if {Q η (f)} η D converges to R(f) for all f M Z.

20 20 SCHERVISH, SEIDENFELD, STERN AND KADANE Lemma 13. Let Z be a set. Let Q = {Q η } η D be a net of coherent previsions on a subset J of M Z. Every cluster point of Q is a coherent prevision on J. Proof. Since every cluster point of every net is the limit of some (possibly different) convergent net, it sufficies to prove the result for limits of convergent nets. Let R be the limit of Q so that, for each J J, lim η Q η (J) = R(J). We show that R is a coherent prevision indirectly. Suppose to the contrary that there are J 1,..., J n J and non-zero real numbers α 1,..., α n and c 1,..., c n and ɛ > 0 such that n α j [J j (z) c j ] ɛ, j=1 for all z Z and such that α j 0 for each j such that R(J j ) = or J j (y) = for some y, and c j = R(J j ) for each j such that R(J j ) <. Since Q η (J j ) converges to R(J j ) for each j, there exists η 0 D such that Q η0 (J j ) R(J j ) ɛ/(n α j ) for all j such that R(J j ) < and Q η0 (J j ) c j for each j such that R(J j ) =. It follows that (12) n α j [J j (z) d j ] ɛ 2, j=1 where d j = Q η0 (J j ) for each j such that R(J j ) < and d j = c j for each j such that R(J j ) =. Since Q η0 (J j ) d j for each j such that Q η0 (J j ) <, the left-hand side of (12) becomes even more negative if we replace each d j by Q η0 (J j ) when R(J j ) = but Q η0 (J j ) <. The resulting inequality implies that Q η0 is incoherent as a prevision for elements of J, a contradiction. Lemma 14. Let Z be a set. Then S Z is dense in Λ Z. Proof. Let λ 0 Λ Z. Let N be a neighborhood of λ 0. Then N contains a basic open set of the form (13) N = {λ Λ Z : λ(f j ) N j, for j = 1,..., n}, where each f j IR Z is bounded below and each N j is a neighborhood of P (f j ) in IR. Define J 1 = {j : λ 0 (f j ) < }, J 2 = {j : λ 0 (f j ) = }.

21 FINITELY-ADDITIVE DECISION THEORY 21 Without loss of generality, we can assume that there exist ɛ j > 0 for each j J 1 and c j for each j J 2 such that N j = (P (f j ) ɛ j, P (f j ) + ɛ j ) for each j J 1, and N j = (c j, ] for each j J 2. Let ɛ = min j J1 ɛ j and c = max j J2 c j. For each j J 2, there exists m j, such that λ 0 (f j m j ) > c + ɛ. Let a = min j inf z f j (z). Let g j = f j a for j J 1 and g j = (f j m j ) a for j J 2. This makes each g j non-negative. We will find a simple probability measure λ N such that λ 0 (g j ) λ N (g j ) < ɛ for all j J 1 and λ N (g j ) > c a for all j J 2. Since f j g j + a, for j J 2, the above clearly implies that λ N N. Let g = n j=1 g j. Note that λ 0 (g) <. Let m be large enough so that λ 0 (g) < λ 0 (gi {g m} ) + ɛ/3, which exists by Lemma 8. Note that Hence n λ 0 (g j I {g m} ) = λ 0 (gi {g m} ) j=1 ɛ 3 > j J 1 > λ 0 (g) ɛ 3, = n λ 0 (g j ) ɛ 3. j=1 [ λ0 (g j ) λ 0 (g j I {g m} ) ]. Since each λ 0 (g j ) λ 0 (g j I {g m} ), we have λ 0 (g j ) < λ 0 (g j I {g m} ) + ɛ/3 for each j. Let A = {z : g(z) m}. Let l = 3m/ɛ. For each j J 1 and k = 1,..., l, let A j,k = A {z : (k 1)ɛ g j (z) < kɛ}. Then, for each j J 1, A j,1,..., A j,l partitions A. Let B 1,..., B r be the elements of the common refinement of these partitions that satisfy λ 0 (B k ) > 0 for each k. If z k B k for all k, then r λ 0 (B k )g j (z k ) λ 0 (g j I A ) < 2ɛ 3, k=1 for all j. Let z 0 A and B 0 = A C. For E Z, define r λ N (E) = λ 0 (B k )I E (z k ). Then λ N (g j ) = k=0 r λ 0 (B k )g j (z k ) + λ 0 (A C )g j (z 0 ). k=1

22 22 SCHERVISH, SEIDENFELD, STERN AND KADANE Since g j (z 0 ) 0, for each j J 2, we have λ N (g j ) λ 0 (g j I A ) 2ɛ 3 > λ 0(g j ) ɛ > c + ɛ a c a. Because z 0 A, we have 0 g j (z 0 ) g(z 0 ) m. Hence, for j J 1, It follows that, for j J 1, g j (z 0 )λ 0 (A C ) λ 0 (gi A C ) ɛ 3. λ 0 (g j ) ɛ < λ 0 (g j I A ) 2ɛ 3 λ N(g j ) λ 0 (g j I A )+ 2ɛ 3 +g j(z 0 )λ 0 (A C ) λ 0 (g j )+ 2ɛ 3. So λ N (g j ) λ 0 (g j ) < ɛ for all j J 1 and λ N (g j ) > c a for all j J 2 and λ N N. Corollary 2 and Lemma 14 combine to provide the following representation. Corollary 3. Let Z be a set. A coherent prevision λ is in Λ Z if and only if there exists a net {λ η } η D in S Z of simple previsions and a coherent prevision P Λ D corresponding to an ultrafilter on D containing all tails of D such that λ(f) = λ η (f)p (dη), for all f M Z. D 4.4. Order Matters. The following is as close as we come to a Fubini/Tonelli theorem involving finitely-additive integrals. Lemma 15. Let X and Y be sets. Let P be the finitely-additive probability corresponding to an ultrafilter U of subsets of Y. Let Q be a countablyadditive discrete probability on X. Then P [Q] = Q[P ] for bounded functions. Proof. Let A X Y. For each x X, let A x = {y Y : (x, y) A}. For each y Y, let A y = {x X : (x, y) A}. For each x X, P (A x ) = 1 or 0 according as A x U or A x U. Let C A = {x : A x U}. Then Q[P ](A) = Q(C A ). For each y Y, let f A (y) = Q(A y ). Lemma 12 says that P [Q](A) equals the common value of sup B U inf y B f A (y) = inf B U sup y B f A (y). For each ɛ > 0, define D A,ɛ = {y : f A (y) (P [Q](A) ɛ, P [Q](A)+ɛ)}. Note that D A,ɛ U. The preceding analysis applies to every subset A of X Y.

23 FINITELY-ADDITIVE DECISION THEORY 23 We would like to prove that, for every ɛ > 0, (14) {y : Q(C A ) ɛ f A (y) Q(C A ) + ɛ} U. This would imply that P [Q](A) Q(C A ) ɛ for all ɛ > 0, hence P [Q](A) = Q(C A ) = Q[P ](A). There exists a countable subset {z 1, z 2,...} of X such that j=1 Q({z j}) = 1. Let {x 1,..., x n } be a large enough subset of C A so that n j=1 Q({x j}) > Q(C A ) ɛ. Since each A x U, n A n j=1 A x j U and for each y n A, {x 1,..., x n } A y. It follows that f A (y) Q(A) ɛ for every y n A. Apply the argument immediately above to A C to conclude that f A C (y) Q(CA C) ɛ for all y in an ultrafilter set that we will call n A. Notice that (A C ) x = (A x ) C for all x X. It follows that C A C = CA C and f A C (y) = 1 f A(y) for all y Y. Hence, Q(C A C ) = 1 Q(C A ), and f A (y) Q(C A ) + ɛ for every y n A. The set n A n A U is then a subset of the set defined in (14). Here is an example to show why Lemma 15 has the condition that Q be a discrete countably-additive probability rather than a general countablyadditive probability. Example 4. Let X = {0, 1}, the set of countable binary sequences. Let Q 0 be the distribution a sequence X = {X n } n=1 of independent Bernoulli random variables each with Pr(X n = 1) = 1/2. Let Y be the positive integers, and let P be the finitely-additive probability corresponding to a non-principal ultrafilter U of subsets of the integers. Let E X Y be defined as follows: For each y Y, let E y = {x : x y = 1}, i.e., the set of sequences in which the yth coordinate is 1. Each E y is a measurable set. Set E = y=1 Ey, which is also measurable, and each E y is the y- section of E. For each x X, the x-section of E is E x = {y : x y = 1}, i.e. the subscripts of those terms in the sequence x that equal 1. We will also need to use the set A = {x : E x U}, which is not measurable. To see this, suppose that A is measurable. Then A is in the tail σ-field of the sequence X. By the Kolmogorov 0-1 law, Q 0 (A) {0, 1}. Similarly, A C is in the tail σ-field, so Q 0 (A C ) {0, 1}. But A C = {x : Ex C U} where Ex C = {y : x y = 0}. By the inherent symmetry in the distribution of X, we must have Q 0 (A) = Q 0 (A C ). But we can t have Q 0 (A) = Q 0 (A C ) = 0, and we can t have Q 0 (A) = Q 0 (A C ) = 1, so A is not measurable. It follows that the inner and outer measures of A are not equal, so there is a number c 1/2 that is between the inner and outer measures. By symmetry the interval between the inner and outer measures is symmetric around 1/2, so we can take c > 1/2. Extend Q 0 to the measure Q on the σ-field generated by A and the original domain of Q 0 with Q(A) = c.

24 24 SCHERVISH, SEIDENFELD, STERN AND KADANE The remainder of the example is devoted to showing that P [Q](E) Q[P ](E). First, note that Q(E y ) = Q(X y = 1) = 1/2 for all y Y, hence P [Q](E) = Q(E y )P (dy) = 1 2. Next, note that P (E x ) = I A (x), so Q[P ](E) = P (E x )Q(dx) = X Y X I A (x)q(dx) = Q(A) = c > 1 2. Lemma 16. Under the conditions of Lemma 15, the minimum extension of the common prevision Q[P ] = P [Q] is the iterated integral Q[P ] of the minimum extensions. Proof. The minimum extension of Q[P ] is Q (f) = lim [f(x, y) n]p (dy)q(dx) n X Y = lim [f(x, y) n]p (dy)q(dx) X n Y = f(x, y)p (dy)q(dx), X Y where the second equality follows from monotone convergence and the fact that Q is countably additive. 5. Separation of Convex Sets. The following definition extends some common terms to deal with IR-valued functions. Definition 14. and f IR Z, define Let Z be a set. Let F IR Z be convex. For each p F A p,f,f = {a (0, ) : p + f/a F }. We say that p is an internal point of F if for every bounded f, A p,f,f. If 0 is an internal point of F, define t F (f) = inf A 0,f,F. We call t F the support function of F.

25 FINITELY-ADDITIVE DECISION THEORY 25 Example 5. Let g IR Z and F = {f IR Z : f(z) < g(z), for all z}. Then F is convex and g ɛ is an internal point for each ɛ > 0. In fact, for every ɛ > 0 and every bounded h ɛ, g h is an internal point of F. Functions that get arbitrarily close to g are not internal. Lemma 17. Let g and k be real-valued functions. Let F be a convex set with g as an internal point. Let F be another convex set that contains k. Suppose that f f is well defined for every f F and every f F. Then k g is internal to F F. Proof. Let h be a bounded function. We need to show that there exists a > 0 such that k g + h/a F F. Since g is internal to F and h is bounded, there exists a > 0 such that g + ( h)/a F. Then k [g + ( h)/a] = k g + h/a F F. (Note that none of the arithmetic in this proof involved infinite values.) Lemma 18. Let F be a convex set with 0 as an internal point. Then 1. if f F, t F (f) [0, 1], 2. if f F, t F (f) [1, ], 3. t F (f 1 + f 2 ) t F (f 1 ) + t F (f 2 ) for all f 1, f 2, 4. t F (αf) = αt F (f) for all f and all α > 0, and 5. F F = {f : t F (f) < and t F ( f) < } is a linear space that contains all bounded functions. Proof. Part (i): Let f F. Since f/1 = f, we see that 1 A 0,f,F and t F (f) 1. Part (ii): Let f F. If A 0,f,F =, then t F (f) = > 1. For the rest of the proof of part (ii), suppose that A 0,f,F. Note that f = a(f/a) + (1 a)0 for all a (0, 1]. So f/a F for a (0, 1] and t F (f) 1. Part (iii): The inequality holds if either t F (f 1 ) = or t F (f 2 ) =. For the rest of the proof of part (iii), assume that α j = t F (f j ) < for both j = 1, 2. Let ɛ > 0. Then f j /(α j + ɛ) F for j = 1, 2. Since F is convex α 1 + ɛ f 1 α 1 + α 2 + 2ɛ α 1 + ɛ + α 2 + ɛ f 2 α 1 + α 2 + 2ɛ α 2 + ɛ = f 1 + f 2 α 1 + α 2 + 2ɛ, is in F. Hence t F (f 1 +f 2 ) α 1 +α 2 +2ɛ for all ɛ, which implies t F (f 1 +f 2 ) α 1 + α 2. Part (iv): Let α > 0. For every a > 0 and every f, αf/(αa) F if and only if f/a F, so that A 0,αf,F = αa 0,f,F for all f. Hence t F (αf) = αt F (f).