EXTENSION OF LYAPUNOV S CONVEXITY THEOREM TO SUBRANGES

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S (XX) EXTENSION OF LYAPUNOV S CONVEXITY THEOREM TO SUBRANGES PENG DAI AND EUGENE A. FEINBERG (Communicated by Mark M. Meerschaert) Abstract. Consider a measurable space with a finite vector measure. This measure defines a mapping of the σ-field into a Euclidean space. According to Lyapunov s convexity theorem, the range of this mapping is compact and, if the measure is atomless, this range is convex. Similar ranges are also defined for measurable subsets of the space. We show that the union of the ranges of all subsets having the same given vector measure is also compact and, if the measure is atomless, it is convex. We further provide a geometrically constructed convex compactum in the Euclidean space that contains this union. The equality of these two sets, that holds for two-dimensional measures, can be violated in higher dimensions. 1. Introduction Let (X, F) be a measurable space and µ = (µ 1,..., µ m ), m = 1, 2,..., be a finite vector measure on it. For each Y F consider the range R µ (Y ) = {µ (Z) : Z F, Z Y } R m of the vector measures of all its measurable subsets Z. Lyapunov s convexity theorem [11] states that the range R µ (X) is compact and furthermore, if µ is atomless, this range is convex. Of course, this is also true for any Y F. Let Sµ(X) p be the set of all measurable subsets of X with the vector measure p R µ (X), Sµ p (X) = {Y F : µ (Y ) = p}. Of course, Sµ p (X) =, if p / R µ (X). For p R m consider the union of the ranges of all subsets of X with the vector measure p, Rµ p (X) = R µ (Y ). Y S p µ(x) In particular, R p µ (X) =, if p / R µ (X), and R µ(x) µ (X) = R µ (X). Since the relation Y 1 = Y 2 (µ-everywhere) is an equivalence relation on F, it partitions any subset of F into equivalence classes. For an atomless µ, Lyapunov [11, Theorem III] proved that: (i) S p µ (X) consists of one equivalence class if and only if p is an extreme point of R µ (X), and (ii) if p R µ (X) is not an extreme Received by the editors February 28, Mathematics Subject Classification. Primary 60A10, 28A10. Key words and phrases. Atomless vector measure, Lyapunov s convexity theorem, purification of transition probabilities. This research was partially supported by NSF grants CMMI and CMMI c XXXX American Mathematical Society

2 2 PENG DAI AND EUGENE A. FEINBERG point of R p µ(x), then the set of equivalence classes in S p µ (X) has cardinality of the continuum. In general, a union of an infinite number of compact convex sets may be neither closed nor convex. As follows from Dai and Feinberg [1], the set R p µ(x) is a convex compactum, if m = 2 and µ is atomless. This fact follows from stronger results that hold for m = 2. For m = 2 and atomless µ, Dai and Feinberg [1, Theorem 2.3] showed that there exists a set Z S p µ (X), called a maximal subset, such that (1.1) R µ (Z ) = R p µ(x), and, in addition, the following equality holds (1.2) R p µ(x) = Q p µ(x), where Q p µ(x) is the intersection of R µ (X) with its shift by a vector (µ(x) p), (1.3) Q p µ (X) = (R µ (X) {µ (X) p}) R µ (X), with S 1 S 2 = {q r : q S 1, r S 2 } for S 1, S 2 R m. In particular, R µ (X) {r} is a parallel shift of R µ (X) by r R m. Examples 3.3 and 3.4 below demonstrate that equalities (1.1) and (1.2) may not hold when µ is not atomless even for m = 1. Each of equalities (1.1) and (1.2) implies that R p µ(x) is convex and compact. However, [1, Example 4.2] demonstrates that a maximal set Z may not exist for an atomless vector measure µ when m > 2. In this paper, we prove (Theorem 2.1) that for any natural number m the set R p µ(x) is compact and, if µ is atomless, this set is convex. This is a generalization of Lyapunov s convexity theorem, which is a particular case of this statement for p = µ(x). We also prove that R p µ(x) Q p µ(x) (Theorem 2.2). Example 3.1 demonstrates that it is possible that equality (1.2) may not hold when m > 2 and µ is atomless. For a countable set A, consider a set {p a : a A} of m-dimensional vectors. The following condition (1.4) (i) a A p a = µ(x), and (ii) a B p a R µ (X) for any finite subset B A. is obviously necessary for the existence of a partition {X a : X a F, a A} of the set X such that µ(x a ) = p a for all a A. According to Dai and Feinberg [1, Theorem 2.5], (1.4) is necessary and sufficient condition for the existence such a partition, when m = 2, µ is atomless, and A is countable. Example 3.2 below demonstrates that this condition is not sufficient for the existence of such a partition for an atomless µ, when m > 2 and A consists of more than two points. By using Lyapunov s theorem, Dvoretzky, Wald, and Wolfowitz [2, 3] proved the existence of the described partition when µ is atomless, A is finite, and (1.5) p a = π(a x)µ(dx), a A. X for some transition probability probability π from X to A. Edwards [5, Theorem 4.5] generalized this result to countable A. Khan and Rath [8, Theorem 2] gave another proof of this generalization. The existence of such a partition is equivalent to the existence of a measurable mapping ϕ : (X, F) A such that for any B A (1.6) I{ϕ(x) B}µ i (dx) = π 2 (B x) µ i (dx), i = 1,..., m. X X

3 EXTENSION OF LYAPUNOV S CONVEXITY THEOREM TO SUBRANGES 3 Consider a measurable space (A, A). According to the contemporary terminology, a transition probability π can be purified if there exists a measurable function ϕ : (X, F) (A, A) satisfying (1.6) for all B A. Loeb and Sun [9, Example 2.7] constructed an elegant example when a transition probability cannot be purified for m = 2, X = [0, 1], A = [ 1, 1], and atomless µ. However, purification holds for nowhere countably generated measurable spaces (X, F, µ), also called saturated. Loeb and Sun [9] discovered this for nonatomic Loeb measure spaces, Podczeck [12] extended this result to saturated spaces by using specially developed functional analysis techniques, and Loeb and Sun [10] showed that, by using the methods from Hoover and Keisler [6], purification can be easily extended from Loeb s measure spaces to saturated spaces; see also Keisler and Sun [7] for properties of saturated spaces. 2. Main results Theorem 2.1. For any vector p R µ (X), the set R p µ(x) is compact and, in addition, if the vector measure µ is atomless, this set is convex. Proof. We say that a partition is measurable, if all its elements are measurable sets. Consider the set V µ,3 (X) = {(µ(s 1 ), µ(s 2 ), µ(s 3 )) : {S 1, S 2, S 3 } is a measurable partition of X}. According to Dvoretzky, Wald, and Wolfowitz [4, Theorems 1 and 4], V µ,3 (X) is compact and, if µ is atomless, this set is convex. Now let W p µ(x) = {(s 1, s 2, s 3 ) : (s 1, s 2, s 3 ) V µ,3 (X), s 3 = µ(x) p, s 1 + s 2 = p}. This set is compact and, if µ is atomless, it is convex. This is true, because W p µ(x) is an intersection of V µ,3 (X) and two planes in R 3m. These planes are defined by the equations s 3 = µ(x) p and s 1 + s 2 = p respectively. We further define T p µ(x) = { s 1 : (s 1, s 2, s 3 ) W p µ(x) }. Since Tµ(X) p is a projection of Wµ(X), p the set Tµ(X) p is compact and, if µ is atomless, it is convex. The last step of the proof is to show that Tµ(X) p = Rµ(X) p by establishing that (i) Tµ(X) p Rµ(X), p and (ii) Tµ(X) p Rµ(X). p Indeed, for (i), for any s 1 Tµ(X), p there exists (s 1, s 2, s 3 ) Wµ(X) p or equivalently there exists a measurable partition {S 1, S 2, S 3 } of X such that µ(s 3 ) = µ(x) p and µ(s 1 ) + µ(s 2 ) = p. Let Z = S 1 S 2. Then µ(z) = p, s 1 R µ (Z), and thus s 1 Rµ(X). p For (ii), for any s 1 Rµ(X), p there exists a set Z F, such that µ(z) = p and s 1 R µ (Z), which further implies that there exists a measurable subset S 1 of Z such that µ(s 1 ) = s 1. Let S 2 = Z \ S 1 and S 3 = X \ Z. Then µ(s 1 ) + µ(s 2 ) = p and µ(s 3 ) = µ(x) p, which further implies that (s 1, µ(s 2 ), µ(s 3 )) Wµ(X). p Thus s 1 Tµ(X). p Theorem 2.2. R p µ(x) Q p µ(x) for any vector p R µ (X). Recall that R p µ(x) = Q p µ(x) when m = 2 and µ is atomless; see (1.2). The proof of Theorem 2.2 uses the following lemma. Lemma 2.3. ([1, Lemma 3.3]) For any vector p R(X), each of the sets R p µ (X) and Q p µ (X) is centrally symmetric with the center 1 2 p.

4 4 PENG DAI AND EUGENE A. FEINBERG Though it is assumed in [1] that the measure µ is atomless, this assumption is not used in the proofs of Lemmas in [1]. Proof of Theorem 2.2. Let q Rµ p (X). Since Rµ(X) p R µ (X), then q R µ (X). Furthermore, in view of Lemma 2.3, p q Rµ p (X). Therefore, p q R µ (X). Since R µ (X) is centrally symmetric with the center 1 2 µ(x), then R µ(x) = {µ(x)} R µ (X) and p q R µ (X) = {µ(x)} R µ (X). Therefore, q R µ (X) {µ (X) p}. As follows from the definition of Q p µ (X) in (1.3), q Q p µ (X). 3. Counterexamples The first example shows that equality (1.2) may not hold when µ is atomless and m > 2. In particular, the inclusion in Theorem 2.2 cannot be substituted with the equality. Example 3.1. Consider the measure space (X, B, µ), where X = [0, 6], B is the Borel σ-field on X, and µ(dx) = (µ 1, µ 2, µ 3 ) (dx) = (f 1 (x), f 2 (x), f 3 (x)) dx, where 30 x [0, 1), 40 x [0, 1), 10 x [0, 1), 40 x [1, 2), 10 x [1, 2), 20 x [1, 3), f 1 (x) = 10 x [2, 4), f 2 (x) = 20 x [2, 4), f 3 (x) = 30 x [3, 4), 15 x [4, 5), 10 x [4, 5), 20 x [4, 5), 5 x [5, 6]; 30 x [5, 6]; 25 x [5, 6]. These density functions are plotted in Fig Note that µ(x) = (110, 130, 125) Figure 1. Density functions of the vector measure in Example 3.1. and { 6 } R µ (X) = α i p i : α i [0, 1], i = 1,..., 6

5 EXTENSION OF LYAPUNOV S CONVEXITY THEOREM TO SUBRANGES 5 is a zonotope, where p 1 = µ([0, 1)) = (30, 40, 10), p 2 = µ([1, 2)) = (40, 10, 20), p 3 = µ([2, 3)) = (10, 20, 20), p 4 = µ([3, 4)) = (10, 20, 30), p 5 = µ([4, 5)) = (15, 10, 20), p 6 = µ([5, 6)) = (5, 30, 25). Let p = p 1 +p 2 +p 3 = (80, 70, 50). Observe that p is an extreme point of R µ (X). Indeed, consider the vector d = ( 7 5, 1, 5) 8 and the linear function ld (α) defined for all α = (α 1,..., α 6 ) R 6 by the scalar product ( 6 ) l d (α) = d α i p i = 6 α i (d p i ) = 66α α 2 + 2α 3 14α 4 α 5 3α 6. For α [0, 1] 6, this function achieves maximum at the unique point α = (1, 1, 1, 0, 0, 0), and l d (α ) = = 102. In addition, 6 α i pi = p. So, d r for all r R µ (X) and the equality holds if and only if r = p. Thus, d r 102 = 0 is a supporting hyperplane of the convex polytope R µ (X), and the intersection of the polytope and hyperplane consists of the single point p. This implies that p is an extreme point of R µ (X). According to the definition of R µ (X), for p R µ (X) there exists a measurable subset Z F such that µ(z) = p and, according to [11, Theorem III] described in Section 1, since p is extreme, such Z is unique up to null sets. In particular, p = µ(z) for Z = [0, 3]. Thus, { 3 } Rµ(X) p = R µ (Z) = α i p i : α i [0, 1], i = 1, 2, 3. Choose q = (56, 29, 31) and observe that q / Rµ(X). p Indeed, q Rµ(X) p if and only if there exist α 1, α 2, α 3 [0, 1], such that 3 α ip i = q, which is equivalent to (3.1) α 1 (30, 40, 10) + α 2 (40, 10, 20) + α 3 (10, 20, 20) = (56, 29, 31), but the only solution to the linear system of equations (3.1) is α 1 = 3 10, α 2 = 11 10, α 3 = 3 10, where α 2 / [0, 1]. On the other hand, q Q p µ(x), because: (i) q R µ (X), and (ii) q R µ (X) {µ(x) p}. Indeed, (i) holds since, for Z 1 = [ ) [ ) [ ) [ 0, , , , ), µ (Z 1 ) = (30, 40, 10) + (40, 10, 20) (10, 20, 20) + (15, 10, 20) = (56, 29, 31) = q. Notice that (ii) is equivalent to q + µ(x) p R µ (X), where q + µ(x) p = (56, 29, 31)+(110, 130, 125) (80, 70, 50) = (86, 89, 106). Let Z 2 = [ ) [ 0, , )

6 6 PENG DAI AND EUGENE A. FEINBERG [ ) [ 2, [3, 5) 5, 5 3 5). Then µ (Z 2 ) = (30, 40, 10) + (40, 10, 20) + (10, 20, 20) (10, 20, 30) + 1 (15, 10, 20) + 3 (5, 30, 25) 5 = (86, 89, 106) = q + µ(x) p. Thus (ii) holds too, and R p µ(x) Q p µ(x). The following example demonstrates that the necessary condition (1.4) for the existence of a measurable partition {X a : a A} with µ(x a ) = p a, a A, is not sufficient for an atomless measure µ when m > 2. In this example, A consists of three points. According to [1, Theorem 2.5], this condition is necessary and sufficient when m = 2, A is countable, and µ is atomless. If A consists of two points, say a and b, and p a R µ (X), p b = µ(x) p a, then the partition {X a, X b } always exists with X a selected as any X a F satisfying µ(x a ) = p a and with X b = X \ X a. Example 3.2. Consider the measure space (X, B, µ) defined in Example 3.1. Let p 1 = (56, 29, 31), p 2 = (24, 41, 19), p 3 = (30, 60, 75), and A = {1, 2, 3}. Then p 1 + p 2 + p 3 = µ(x). We further observe that: (i) p 1 is the vector q from Example 3.1, so p 1 R µ (X) and therefore p 2 + p 3 = µ(x) p 1 R µ (X); (ii) p 1 + p 3 is the vector q + µ(x) p from Example 3.1, so p 1 + p 3 R µ (X) and therefore p 2 = µ(x) p 1 p 3 R µ (X); (iii) p 1 + p 2 is the vector p from Example 3.1, so p 1 + p 2 R µ (X) and therefore p 3 = µ(x) p 1 p 2 R µ (X). Thus, the vectors p a, a A, satisfy (1.4). If there exists a partition {X a B : a A} of X with µ(x a ) = p a for all a A, let Y = X 1 X 2. Since X 1 X 2 =, µ(x 1 ) = p 1 = q, and µ(y ) = p 1 +p 2 = p, then q Rµ(X). p However, according to Example 3.1, q / Rµ(X). p This contradiction implies that a partition {X a B : a A} of X, with µ(x a ) = p a for all a A, does not exist. In conclusion, we provide two simple examples showing that, if µ is not atomless, then even for m = 1 (and, therefore, for any natural number m) a maximal subset Z, defined in (1.1), may not exist and equality (1.2) may not hold. Example 3.3. Consider the probability space (X, 2 X, µ), where X = {1, 2, 3, 4} and µ({1}) = 0.1, µ({2}) = 0.4, µ({3}) = 0.2, µ({3}) = 0.3. Let p = 0.5. Then S p µ = {{1, 2}, {3, 4}}. In other words, the only subsets that have the measure 0.5 are Z 1 = {1, 2} and Z 2 = {3, 4}. However, R µ (Z 1 ) is not a subset of R µ (Z 2 ) and vice versa. Therefore, a maximal subset does not exist for p = 0.5. Example 3.4. Consider the probability space (X, 2 X, µ), where X = {1, 2, 3} and µ({1}) = 0.1, µ({2}) = 0.55, µ({3}) = The range of µ on X is R µ (X) = {0, 0.1, 0.35, 0.45, 0.55, 0.65, 0.9, 1}. Let p = Then Q p µ = {0, 0.1, 0.45, 0.55} and R p µ = {0.55}. Thus R p µ Q p µ, but R p µ Q p µ.

7 EXTENSION OF LYAPUNOV S CONVEXITY THEOREM TO SUBRANGES 7 References [1] P. Dai and E. A. Feinberg. On maximal ranges of vector measures for subsets and purification of transition probabilities. Proc. Amer. Math. Soc. 139(2011), [2] A. Dvoretzky, A. Wald, and J. Wolfowitz. Elimination of randomization in certain problems of statistics and of the theory of games. Proc. Natl. Acad. Sci. USA 36(1950), [3] A. Dvoretzky, A. Wald, and J. Wolfowitz. Elimination of randomization in certain statistical decision procedures and zero-sum two-person games. Ann. Math. Stat. 22(1951), [4] A. Dvoretzky, A. Wald, and J. Wolfowitz. Relations among certain ranges of vector measures. Pacific J. Math. 1(1951), [5] D. A. Edwards. On a theorem of Dvoretzky, Wald, and Wolfowitz concerning Liapounov measures. Glasg. Math. J. 29(1987), [6] D. N. Hoover and H. J. Keisler. Adapted probability distributions. Trans. Amer. Math. Soc. 286(1984), [7] H. J. Keisler and Y. Sun. Why saturated probability spaces are necessary. Adv. Math. 221(2009), [8] M. A. Khan and K. P. Rath. On games with incomplete information and the Dvoretzky- Wald-Wolfowitz theorem with countable partitions. J. Math. Econom. 45(2009), [9] P. Loeb and Y. Sun. Purification of measure-valued maps. Illinois J. Math. 50(2006), [10] P. Loeb and Y. Sun. Purification and saturation. Proc. Amer. Math. Soc. 137(2009), [11] A. A. Lyapunov. Sur les fonctions-vecteurs complètement additives. Izv. Akad. Nauk SSSR Ser. Mat. 4(1940), (in Russian). [12] K. Podczeck, On purification of measure-valued maps, Econom. Theory 38(2009), Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY address: Peng.Dai@stonybrook.edu Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY address: Eugene.Feinberg@sunysb.edu