ON A CLASS OF PROBABILITY SPACES

Transcription

1 ON A CLASS OF PROBABILITY SPACES DAVID BLACKWELL UNIVERSITY OF CALIFORNIA, BERKELEY 1. Introduction Kolmogorov's model for probability theory [10], in which the basic concept is that of a probability measure P on a Borel field A of subsets of a space S, is by now almost universally considered by workers in probability and statistics to be the appropriate one. In 1948, however, three somewhat disturbing examples were published by Dieudonn6 [2], Andersen and Jessen [1], and Doob [3] and Jessen [9], as follows. A. (Dieudonne). There exist a pair (Q2, A), a probability measure P on A, and a Borel subfield 4 c A for which there is no function Q(w, E) defined for all co E Q, E E A with the following properties: Q is for fixed E an 4-measurable function of co, for fixed co a probability measure on A, and for every A E 4, E E A, we have (1) fjq (w, E) dp (c) =P (A n E). B. (Andersen and Jessen). There exist a sequence of pairs (Qn, B,,) and a function P defined for all sets of u 4,,, where 4,n consists of all subsets of the infinite product space 1X 2X *... in the Borel field determined by sets of the form B1 X... X B,, X Q.+1 X fqn+2x, Bi E Ai, i = 1,, n, such that P is countably additive on each 4,, but not on u 4,. C. (Doob, Jessen). There exist a pair (Q, A), a probability measure P on A, and two real-valued A-measurable functionsf, g on 0 such that (2) P{i: fe F, gegi =P{co: f EF}P{w: geg} holds for every two linear Borel sets F, G but not for every two linear sets F, G for which the three probabilities in (2) are defined. In each case Q is the unit interval, A is the Borel field determined by the Borel sets and one or more sets of outer Lebesgue measure 1 and inner Lebesgue measure 0, and P consists of a suitable extension of Lebesgue measure to A. The fact that A, B, C cannot happen if Q is a Borel set in a Euclidean space and A consists of the Borel subsets of Q is known. For A, the proof was given by Doob [4], for B by Kolmogorov [10], and for C by Hartman [7]. To the extent that A, B, C violate one's intuitive concept of probability, they suggest that the Kolmogorov model is too general, and that a more restricted concept, in which A, B, C cannot happen, is worth considering. In their book [51, Gnedenko and Kolmogorov propose a more restricted concept, that of a perfect probability space, which is a triple (Q, A, P) such that for any real-valued A-measurable function f and any linear E A} E A, there is a Borel set B c A such that set A for which {I:f(Aw) (3) PIw: f (w) E B} =P{ I,: f (w) E A I. This investigation was supported (in part) by a research grant from the National Institutes of Health, Public Health Service. I

2 2 THIIRD BERKELEY SYMPOSIUM: BLACKWELL As noted by Doob [41 (see appendix in [5]) in perfect spaces A, C cannot happen, and it then follows from a theorem of Ionescu Tulcea [8] that B cannot happen in perfect spaces. The concept introduced by the writer here is that of a Lusin space, which is a pair (Q, A) such that (a) A is separable, that is, there is a sequence IB.} of elements of A such that A is the smallest Borel field containing all B,,, and (b) the range of every realvalued B-measurable function f on Q is an analytic set, that is, a set which is the continuous image of the set of irrational numbers. The concept of Lusin space is more restricted than that of perfect space in the sense that if (Q, A) is a Lusin space and P is any probability measure on A, then (Q, B, P) is perfect. It is shown below that for Lusin spaces none of A, B, C can occur. The primary property of Lusin spaces which ensures this regularity, and which fails for the example of A, B, C mentioned above, is that the only events whose occurrence or nonoccurrence is determined by specifying which events in a sequence E1, E2,. * occur are the events in the Borel field determined by the sequence {En,,. This property permits the identification of the concepts, for real-valued A-measurable functions f, g, "f is a function of g" and "f is a Baire function of g," the nonequivalence of which in general is a technical nuisance to say the least. 2. Preliminaries In this section we list some definitions and some known properties of analytic sets to be used in later sections. If M is a metric space, the sets in the smallest Borel field containing all open sets will be called the Borel sets of M. A Borel field A of subets of a space Q will be called separable if there is a sequence {B.} of sets in A such that A is the smallest Borel field containing all B,,. Thus if Q2 is a separable metric space, the class of Borel sets is a separable Borel field, though not conversely. If A is a separable Borel field of subsets of Q2 and {B,,I is a sequence determining A, the sets of the form n C,, where each C. is either B. or Q- B., are called the atoms of A. Any two nonidentical atoms are disjoint and every set in A is a union of atoms, so that the class of atoms of A is independent of the particular sequence {B,. A metric space A will be called analytic if A is the continuous image of the set of irrational numbers. We shall use the following properties of analytic sets, due to Lusin [11]. I. If { A,,} is a sequence of analytic sets in a metric space M, then u A,,, n A,, if nonempty, the product space A1 X A2, and the infinite product space A1 X A2 X... are analytic sets. II. If A is analytic, so is every Borel subset of A. III. Every Borel set of Euclidean n-space is analytic. IV. If A, B are disjoint analytic subsets of a metric space M, there is a Borel set D of M such that D m A and D is disjoint from B. V. If f is a Borel-measurable mapping of an analytic set A into a separable metric space M, that is, the inverse image of any open set in M is a Borel set of A, thenfa, the range of f, is an analytic set. We shall also use the following property pointed out to the writer by A. P. Morse. VI. If P is a probability measure on the Borel sets of a metric space M and A is an analytic subset ofm then for every e > 0 there is a compact C inside A with P(C) > u- e, where ;u = min P(B) as B varies over all Borel sets of M which contain A.

3 ON A CLASS OF PROBABILITY SPACES 3 3. Lusin spaces and analytic sets The main content of theorems 1 and 2 is that, apart from the unessential difference that the atoms of a Lusin space need not be points, Lusin spaces are identical with pairs (2, A) where Q is analytic and A is the class of Borel sets of a. THEoREm 1. If Q is analytic and A is the class of Borel sets of 12, then (Q, A) is a Lusin space. PROOF. Separability of A follows from the separability of Q, and that the range of every A-measurable real-valuedf is analytic is the special case of V with M the real line. TEoRE1m 2. If (Q, A) is a Lusin space whose atoms are points and { En} is any sequence determining A then there is a metric on Q with respect to which Q is an analytic set, A consists of the Borel sets of Q, and every E. is both open and closed. PROOF. Say {En) determines A and let f(w) = I en(w)/3n where en is the characn teristic function of En. Then f is a 1-1 A-measurable map of Q onto an analytic subset A of the line. Let d(wl, CW2) = /lk(coi, W2), where k is the smallest n for which en(wi) 0 en(w2). Then f is bicontinuous between Q and A, and every En is open and closed, since any point not in En has distance at least 1/n from En. Finally, to identify A with the class _% of images of Borel sets of A underf-1, the B-measurability of f implies A v %, and we need only show E,n E._ to conclude _% n A. Since En is the image under fi of the set of numbers in A whose nth ternary digit is 1, the proof is complete. 4. Set theoretic properties of Lusin spaces THEoREm 3. If (Q, A) is a Lusin space, e is a separable Borel field of A-sets and A E is a union of atoms of e, then A E e. PROOF. Say {C.} determines e and let f(w) = S cn(w)/3n. Then f maps e-atoms n into points, and different e-atoms into different points. ThenfA and f(q2- A) are disjoint analytic linear sets, so that from property IV there is a linear Borel set D such that D : fa and D is disjoint from f(q- A). Consequently f-1d = A, so that, since f is e-measurable, A E e. COROLLARY 1. If (Q, A) is a Lusin space, two separable Borel fields of A-sets with the same atoms are identical. COROLLARY 2. Let (2, B) be a Lusin space and letf map Q onto an arbitrary space Z. If there is a separable Borel field e C A whose atoms are the sets f l(z), z E Z, then e is identical with the class of all sets in A of the form f-1d, D c Z, and (Z, %) is a Lusin space, where _% consists of all D E.a for which f-'d E A. PROOF. The mapping fp is a 1-1 mapping between the points z E Z and the atoms of e. Thus every C E e has the form fld for some D c Z. Conversely if A = fld for some D c Z, A is a union of atoms of e, so that, from the theorem, A E A implies A E e. Thus D E % if and only iff-d E e. It follows that if {C"} generates e, then {fcn7 generates.%, so that _% is separable. Finally, if h is any real-valued.%-measurable function on Z, then hf is a e-measurable function on Q whose range is the same as the range of h. Since (1, A) is a Lusin space, this range is analytic and the proof is complete. COROLLARY 3. Let (1, A) be a Lusin space, let f map Q onto an arbitrary space Z, and denote by J the Borel field of all Z-sets Sfor which ft1s E A. For any separable _ c g,

4 4 THIRD BERKELEY SYMPOSIUM: BLACKWELL (Z, _%) is a Lusin space. If in addition every S E S is a union of atoms of %, then g = %, so that (Z, 5) is a Lusin space. PROOF. The first conclusion of the corollary follows immediately from the definition of a Lusin space. The second conclusion follows from the first and theorem 3. COROLLARY 4. If (Q, A) is a Lusin space and f is a A-measurable function from Q into a separable metric space M, then for every set A c Mfor whichf-1a E A there is a Borel set B of M such that f-'b = f'a. PROOF. The class e of all sets of the formf-1b, where B is a Borel set of M, is a separable Borel subfield of A. Every set f-1a is a union of atoms of e0 and thus is in e if it is in A. Thus if f-a E A, there is a Borel set B ofm for which f-'b = f-ia. Theorem 3 identifies, for Lusin spaces, the concepts "an event A depends only on events in e" and "A E ex." The following theorem extends this to functions. THEoRE1 4. Let (QT, A) be a Lusin space, let f, g be A-measurablefunctions from Q into separable metric spaces Y, Z, and denote by Bf(A) the class of all sets of the form f-1s (g-1t) where S(T) is a Borel set in Y(Z). (a) If there is a function 0 from Y into Z such that kf = g, then g is measurable with respect to the Borel field off-sets, that is, A0 c Af. (b) If B, c Bf, then there is a Borel-measurable function 4t from Y into Z such that = g- PROOF. (a) Separability of Y implies separability of Af. Every set in A, is a union of atoms of Af, so that (a) follows from theorem 3. The hypothesis that (Q, A) is a Lusin space is not necessary for (b); the proof by Doob [41 for Y, Z Euclidean spaces extends easily to arbitrary separable metric spaces Y, Z. 5. Conditional probability distributions, Kolmogorov extension THEOREm 5. Let (Q, A) be a Lusin space, let P be a probability measure on A, and let,4 be a separable Borel subfield of A. There is a real-valued function Q(co, B) defined for all co E Q, B E A such that (a) for fixed B E A, Q is an 4-measurable function of w, (b) for fixed w, Q is a probability distribution on B, (c) for every A E 4, B AB, f Q(w, B)dP = P(A n B), and (d) there is a set N E4 with P(N) = 0 such that Q(cw, A) = I for o E A, w E N. PROOF. We may suppose that the atoms of A are points. Choose F. E A so that {FR) determines A and a subsequence of {Fn} determines 4, and (theorem 2) metrize Q so that a is analytic, A consists of the Borel sets, and each F. is open and closed. Choose C, compact so that C. c C.+1, P(C,) -- 1, and denote by q, Q the fields determined by F1, F2,. * * and F1, F2 * * *, C1, C2...* respectively. Let Ql(w, B) be defined so as to satisfy (a) and (c) for B E Q7. Since q is countable, there is a set N e 4 with P(N) = 0 such that for co N, (4) Qi is additive and nonnegative on (5) Ql(,w, )= (6) Q1(c, A) =1 for --A E 4, A EQ and (7) Q1(o,Cj )-1 as n--.

5 ON A CLASS OF PROBABILITY SPACES 5 Then Qi is countably additive on q, for if {Ha} is a sequence of disjoint sets of q with u Ha = Q, for every n, since C. is compact and the H. are open and closed, there is an M such that U Hi n C.. Finite additivity of Qi on Q yields, for X E N, Qi(w, C.) _ 1 M co co E Q'(, Hi) s Q'(w, Hi). Letting i n - and using (7) yields j Ql(cw, Hi) _ Additivity yields the reverse inequality, so that, for w E N, Q, is countably additive on 47. For X E N, we define Q(w, B) as the (unique) countably additive extension of Q, from I to B. For X E N, we define Q(w, B) = P(B). Then (b) holds, and the class of sets B for which (a) and (c) hold is a monotone class containing 47, so coincides with A [6]. To verify (d) let A E 4, Xo E A, w E N, and denote by I the 4-atom containing w. Then I c A and, since a subsequence of IF I determines 4, there is a sequence Jn E 47 for which nj. = I. From (2),Q(w, J.) = 1 for all n, so that Q(W, I) = 1. This completes the proof. THEoREm 6. Let { (Q,, B.) I be a sequence of Lusin spaces, let Q2 be the infinite product... space Q1 X Q2 X and let 4n be the Borel field determined by all sets A1 X X A, X - Qa+1 X Qn+2 X Ai E j. A function P defined on u 4an which is a probability measure on each 4a is countably additive on u a.. PROOF. Let Aa be a decreasing sequence of sets in u 4a with P(An) > 0. We must show that n Aa is not empty. We may suppose that the atoms of B, are points and metrize D. so that it becomes analytic and An consists of the Borel sets of Qn. We may also suppose that Aa E 4q. From property VI there is a set Da E 4n such that Da C A,, P(Da) > P(An) - 5/2a, and D. = Ca X Qn+1 X Qn+2 X* X, where C. is a compact subset of Q1 X *... X.. Since P(Di n... n DN) > P(Al n... n AN) N > 5>> 0,D1 n An DNis nonempty for each N. If ON = (XN1, XN2,*) E D1 n... n DN, we have (xn,, XNk) E Ck for all N _ k, so that there is a subsequence of wn which converges coordinatewise to a point co* = (x*l, x, ), and (x*., x*) E Ck for each k. Thus w* E Dk c Ak for all k and the proof is complete. 6. Independence, perfection THEOREM 7. If (Q2, A) is a Lusin space, P is any probability measure on B, and f, g are any two A-measurable functions from Q into separable metric spaces X, Y such that (8) P{w:f E A, geb} =P{w: f E A}P{gEB} for all Borel sets A, B in X, Y, then (8) holds for all sets A, B in X, Y for which the terms are defined. PROOF. The theorem follows immediately from corollary 4 of theorem 3. THEOREm 8. If (Q, A) is a Lusin space, P is any probability measure on A andf is any A-measurable function from Q into a separable metric space, then inside any A c M for which fla E A there is a Borel set B of M with P(f-1B) = P(ft1A). PROOF. We may suppose A c R, the range of f. If e consists of the Borel sets of R, then (R, C) is a Lusin space and, from corollary 4 of theorem 3, A E e. The function +O(C) = P(flC) is a probability measure on e, and from property VI there is inside A a union B of compact sets with +(B) = O(A).

6 6 THIRD BERKELEY SYMPOSIUM: BLACKWELL 7. Some unsolved problems Problem 1. If B is a separable Borel field of subsets of a space Q such that every separable Borel subfield of B with the same atoms is identical with 3, is (2, R) a Lusin space? Problem 2. If B is a separable Borel field of subsets of a space Q such that (Q, /, P) is perfect for every probability measure P on B, is (Q, 8) a Lusin space? Problem 3. Can the exceptional set N be eliminated from (d) of theorem 5? That is, given an analytic set N and a separable Borel subfield 4 of the class / of Borel sets of N, does there exist a function Q(co, B) defined for all co E N and all B E B which (i) for fixed B is an 4-measurable function of w, (ii) for fixed w is a probability distribution on B and (iii) for which Q(w, A) = I for all A E, coe A? REFERENCES [1] E. SPARRE ANDERSEN and BORGE JESSEN, "On the introduction of measures in infinite product sets," Danske Vid. Selsk. Mat.-Fys. Medd., Vol. 25, No. 4 (1948), 8 pp. [2] JEAN DIEUDONNE, "Sur le th6oreme de Lebesgue-Nikodym. III," Ann. Univ. Grenoble, Vol. 23 (1948), pp [3] J. L. DOOB, "On a problem of Marczewski," Colloquium Mathematicum, Vol. 1 (1948), pp [4], Stochastic Processes, New York, John Wiley and Sons, [5] B. V. GNEDENKo and A. N. KOLMOGOROV, Limit Distributions for Sums of Independent Random Variables. Translated by K. L. Chung with an appendix by J. L. Doob, Cambridge, Addison- Wesley, [61 P. L. HALMOS, Measure Theory, New York, D. Van Nostrand, [7] S. HARTMAN, "Sur deux notions de fonctions independantes," Colloquium Mathematicum, Vol. 1 (1948), pp [8] C. T. IONEScU TULCEA, "Measures dans les espaces produits," Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., Vol. 7, Series 8 (1949), pp [9] B0RGE JESSEN, "On two notions of independent functions," Colloquium Mathematicum, Vol. 1 (1948), pp [10] A. N. KOLMOGOROv, Grundbegri.ffe der Wahrscheinlichkeitsrechnung, Ergebnisse Mathematik, Berlin, Springer, [11] NicoLAs LusIN, Lecons sur les Ensembles Analytiques et leurs Applications, Paris, Gauthier-Villars, 1930.