PROBABILITY MEASURES ON A HILBERT SPACE
SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF MASTER OF SCIENCE
BY
CONNIE MARKS LEGGETT
DEPARTMENT OF MATHEMATICS
ATLANTA, GEORGIA
July 1976 ACKNOWLEDGMENT
I wish to acknowledge my indebtedness to ray advisor, Dr.
Nazir A. Warsi, for his helpful suggestions, constant encour¬ agement, constructive criticism and patience during the long hours of preparation of this thesis.
C.M.L. ABSTRACT
MATHEMATICS
Leggett, Connie Marks B.S., Albany State College, 1975
Probability Measures On A Hilbert Space
Advisori Dr. Nazir A. Warsi
Thesis dated July 1975
Cylinder Sets which are generated by subsets of the Hilbert
Space are introduced. V/e then consider probability measures
on the sigma-algebra of Cylinder Sets. A random variable, measurable with respect to the Borel sigma-algebra is defined.
Several applications and theorems are discussed. Concepts
are briefly mentioned to suggest aspects of further study as
viewed by cvirrent literature. TABLE OF CONTENTS
Page
ACKNOWLEDGEMENT i
Chapter
I. Preliminaries 1
II. Measures On Cylinder Sets.. 6
Compatibility Condition 6
Necessary Condition to Extend a Measure 7
Extension of Cylinder Measure 7
Characterization of Countably Additive Cylinder
Measiure 7
Characteristic Fvinstions and Countable
Additivity 9
III. Random Variables l4
IV. Applications 17
Example of Cylinder Set Measure 17
Suggestions For Further Study 19 CHAPTER I
PRELIMINARIES
Cylinder Sets.
Definition 1.1. Let H be separable Hilbert space.
For a finite dimensional subspace M in H, a cylinder set is any set of the formj
C = B + M"*" where B is a Borel subset of M and M'*"is the orthogonal complement of M. The plus sign between two sets will denote direct sum.
Definition 1.2. The Borel set B is called the base of the cylinder and M the base space or generating space.
Characterization of Cylinder Sets.
The cylinder sets may be characterized in the following way.
Suppose B is a Borel subset of an n-dimensional
Euclidean space. Also, let X2» distinct elements in H. If =
C = {Yi-Kt * b]
Also, let P be an orthogonal projection of H on M, a subspace of H. For each Y-6C,
Y = P^ + (I-P)Y
1 2
Note I is the identity map and <'X,Y> denotes the inner product of X and Y.
Theorem 1.1» The following properties of cylinder sets hold.
(a) The set-theoretic complement C of a cylinder set C is a cylinder set.
(b) If and are cylinder sets then C^nC2 is a cylinder set.
(c) If and are cylinder sets then ^^^^2 ^ cylinder set.
Proof
(a) Let C be a cylinder set with base B then C = B + M'^.
The claim is that C = B' + M"*”.
If X-€C' then X = b + h with h in M and b not in B.
Thus b is an element of B' and X lies in B' + M"^.
Conversely, let X belong to B' + M“K Then X = b + h where b is in B' and h lies in M"^. This implies X is not an element of C. Hence X is an element of C*.
Therefore, C = B' + M'S Since B' is also a Borel subset of M, C is a cylinder set.
(b) Let and C2 be cylinder sets with base B^ and B2 respectively in subspace and H2 respectively. If is spanned by then it can be verified that =
H^nH2 • Now = B^ + = B^ + (H^ - also
G2 = B2 + H2 = B2 + (H^ ” . 3 J- i- X Since 4- ) and
H2 = + (Hg - ),
- H^OH^ and - H^CH^. Let'S'^ = + (H^ - ) and ^2 ^2 “ ^3^ then and ^2 Borel sets. Hence
G^n C2 = Bj_4-(H^ - ) + H2nB2 + (H2 - H^ ) + = \ + H^nB2 +
= B'^n'B2 + . Thus C^n C2 is a cylinder set. (c) Let and C2 Le cylinder sets then from (a) C^' and C2' are also cylinder sets. Since from (b) Cj^nC2 is a cylinder set, (0^002)' = C^'UC^' is a cylinder set.
Hence the class of all cylinder sets^form a "field".
Definition 1.3» The smallest sigma-algebra of sets
containing all open sets in H is called sigma-algebra of Borel sets in H. Denote this sigma-algebra by^. Theorem 1.2* The class of Borel sets^ is also the smallest sigma-algebra containing all cylinder sets.
Proof
First show that every cylinder set is a Borel set.
Let C be a cylinder set with base B in M. Suppose B
is a closed set in M, then G = B + M"*" is closed. This
happens as a result of being a closed linear subspace. 4
Let Q the smallest sigma-algebra containing all cylinder sets with closed sets as bases in the (fixed) base space M. Then y^j,Cy§'that ieis a sub sigma-algebra of Hence are Borel sets in H. Borel sets in M are the members of the smallest sigma-algebra generated by closed sets in M. Thus cylinder sets with Borel sets as bases are
Borel sets in H. Therefore, the smallest sigma-algebra containing all cylinder sets must be contained in^^ that is
Now to showConsider A, a Borel set in H.
Show that A is a member of the smallest sigma-algebra generated by the cylinder sets.
Consider the special case where A is a closed sphere,
A ={xt II X-Xo II ^ - m(.
Let be a complete orthonormal system then
A^ = [xi ^(^X - is a cylinder set for each n and A - Hence a closed sphere is a member
Since H is sepsirable, every open set is the countable union of closed spheres. For U an open set in H, let be a coimtably dense set in H. Let be the subsequence contained in U. Then for each Zj^., let S(Zj^.) = ^ S (Zj^, r^), where {is the sequence of all rational numbers.
S(Zj^, r^) = {xj II - X ll ^ r^i where only those S(zj^, r^) contained in U are included. 5
Hence S(Zj^) is an element and U is contained in ys(Zj^). If y is in U then there exists a closed sphere with rational radius such that S(y,r) is contained in U. Now
S(y,r) must contain a subsequence ^ converging to y. Then y is an element of r^) which is contained in
S(y,r) and S(y,r) is contained in U. This happens for some sufficiently small r^^. Also, Vs(Zj^) is contained in U or U = ys(Zj^). Therefore U is a member of But since is a sigma-algebra generated by open sets, Since is contained in y^andcontains/'^, it follows that^^ =/3^. CHAPTER II
MEASURES ON CYLINDER SETS
Introduction. The first two ingredients of a
"probability” spacei H, the Hilbert Space and the Borel sigma-algebra were discussed in the previous chapter.
A probability on B is now induced. Such a measure denoted
t will be referred to as "cylinder set" measure.
Definition 2.1. Let Z be a cylinder set with base B in M. Then we define /^(Z) =t/(B) where is a probability measure on the sigma-algebra of
Borel subsets of M.
Specifically, for a disjoint set of cylinder sets
( Zjj.\with corresponding base sets in common base space M,
k=l
Compatibility Condition.
In order thatbe well-defined, it is necessary that ifZ = B'fM =B + Hp + (Hjjj + Hp) where P is orthogonal to M then = + where is the Borel v^„(B)m' ' m+p (B H_)p' ra+p measure on M + P.
6 7
Necessary Condition to Extend A Measure.
Lety6<-be a cylinder measure. It can be extended to be countably additive on the sigma-algebra of bored sets generated by if and only if for any cylinder set Z such oD that ^ Z n=l n* where Z are cylinder sets. Then < n=l/^^^n^* This is equivalent to saying that if H= 2^ is a sequence of mutally disjoint cylinder sets.
Then ao 1§1
Extension of Cylinder Measure.
In order to extend define an "outer" or "caratheodory" measure corresponding to any cylinder set measure.
Definition 2.2. Let F be any subset of H. The extension is defined A oa ^g(F) = Inf k=l/^(2j^)* where f is any sequence of cylinder sets which cover F, that is FCUz^. The Inf is taken over all such sequences. Theorem 2.1. For the extensionand cylinder set Z,
^g(Z) ^(Z). Moreover, if F^F^ then any set sequence covering F^ also covers F^^ so that
Characterization of Countably Additive Cylinder Measure.
Theorem 2.2. A necessary and sufficient condition that a cylinder set raeasu^ey^^be countably additive is that given any ^ 0 there exist a closed boiAnded set K such that y6^g(K) 2 1-e. Proof
Necessity
Suppose^is countably additive. Then^^ is also countably additive on j^.li S(0, N) = ^Xi II Xll"'< N = 1, 2, 3. ••• then S(0, N) is monotonicly increasing and 1= liin^^S(0, N).
Hence given ^2o, there exist N large enough so that
^^S(0, N) ^ 1- e .
Sufficiency
For any ^ ^ 0, there exist a closed bounded set K such
that
/x^(K) 2. 1- t
for lairge X and K C S (0, .
Hence (0, ^ ) 2, 1- e .
Let fbe a sequence of mutally disjoint cylinder sets
such that cl*
For each K, there exist an open cylinder set Zj^ such that given ^ ^ 0,
/^{\) is open in the weak topology. Since S(0, A) is weakly compact. 9
Hence
T. 6/-k+l k=l Therefore <« 7^
kll/^^k) Z k?i/^(Zjj) Z 1-26^ . «e Since ^ is arbitrary, Z 1. also for every N, ^ 1. *i
Therefore = 1.
Characteristic Functions and Countable Additivity.
Definition 2.3» For every probability measureyu, its A characteristic functionyt^(y), y ^H is defined by the formula /l(y) =
Theorem 2»3» The following are basic properties of characteristic functionsi
(a) ^(y) is uniformly continuous in the norm topology, (b) If y^H then~
Proof
(a) For y^, Yl then
= J I 4^
_ 4 sin^^zx, y]L"^2^ *
Let ^ ^ 0 be fixed but arbitrary. Then choose a positive constant K such that /(-(Xj II X//>’K)«c.fc. The following is then obtainedI
I^(y^) “^(y2)| - + 4 J sin^i
•'A-x ^1* ^2**'*^n /^1 ^^2 measures/^ through the cainonical isomorphism corresponding to the basis {. Then the function (yj^»*»*y^) and are the characteristic functions of the finite dimensional measures ofand^2* respectively. They are induced by the projection
(X, , X, (X^, X2. .XJ. Thus =/^ which implies ~/^z' Definition 2.4. Let^be a probability measure such that J"// x// ^ . Then the covariance operator S of Jl/-is the Hermitian operator determined uniquely by the quadratic form
Definition 2.6. hetjldenote the family of all
S-operators. The class of sets { Xj (<'SX,X>) <1, defines a system of neighborhoods at the orgin for a certain topology, which is called the S-topology.
Definition 2.7. A function‘f defined on H is said to be positive definite if and only if for any ^complex numbers a^, any Vpoints yj_, ^ ^ ^
J. J; a-*a^ -fCyx-y;) ^ o.
Theorem 2.4. A fxmction defined on Y is the characteristic fimction of a probability measure /jc if and only if the following conditions holdi
(1) ^(e) = 1, where e is the identity of H. (2) ‘fis continuous. (3) ‘fis positive definite.
Theorem 2.5. In order that function ^(y), y H be the characteristic function of a probability measure^^^ , the following two conditions are necessary and sufficient ii ) ‘f(o) = 1, if(y) is positive definite in y.
(ii ) lf(y) is continuous in the S-topology.
Proof
Sufficiency
Let ^e^l be a (fixed) sequence of orthonormal basis of vectors. Since the S-topology is weaker than the norm topology, it follov/s that ^(y) is continuous in the norm topology. Let {y, • •yn®n^ * 12
Then ^ continuous positive definite function in n-real variables. Hence by Theorem 2.4, there exists a probability measureon the n-dimensional real vector space such that
S^n(yi»y2»* *‘^n^ = / exp (t Vn^ * Let be the subspace spanned by the vectors e-j^, ®2***'®n* '^l^en/c^ can be considered as a measure in Let ^^(A)=^^(A/1H^). Then
^(y) (‘ < X,y > )dyi»= |exp( / (X,y, + .. = f(y, e,t...y„e^). Thus lim/l^(y) =H^(y), y-^ H.
The proof of sufficiency will be complete if the sequence
of measures/x. is compact. It is enough to prove that
J= lim sup (2'ff')"^^'" N+l)/2j". . . j it - R'f n-N+1 times
-yj p X exp(-i .1 y. )dy^...dy =0. 4-/V ‘ Let ^>0 be arbitrary. By hypothesis there exists an
S-operator such that
1-R S (y) ^
for all y such that ("^Styty >) ^2.
Since 1-R*f(y) ^2 for all y •‘•H, it follows that
l-R'f’(y) < (
for all y. Hence, from the definition of J
J 1 lim sup f fe + ? < S,e.. , = ^ •
Letting ^ 0, then J = 0. 13
Necessity
Let K be a compact set such that Kch and^(K) < 2^ . Then
l-R^(y) = J (l-cojcx,y> )d^ (
^Ay,y> = J ^ X,y>^ d^ . If {<^ Ay,y7 ) < 1, then 1-R^ (y)< Since I l-A(y) / ^ < 2 (1 -RyU, (y) ), it follows thatyu(y) is continuous in the S-topology. Since
(y)=/^(y)>'the proof is complete. CHAPTER III
RANDOM VARIABLES
Definition 3«1» Let H be a separable Hilbert Space;
^ , the cylinder sets and j^the Borel field in H. A fxinction, f(w), mapping H into another Hilbert Space
(not necessarily separable) is a basic random variable if
(i) \yi: \ < f (w), B, Borel set in are in ^ , for any number of elements in H.
(ii) y^A-is defined and countably additive on the sub-sigma algebra of inverse images of Borel sets.
Theorem 3.1« Let L denote any bounded linear
transformation mapping H into H. Let m be a fixed element of
H. Then with H as a sample space,
F(w) = L(w) + m
is a basic random variable.
Theorem 3.2. Let C be a cylinder set in H. Then the
inverse image
Iw : L(w) + m -fc C I
is also a cylinder set,
Definition 3.2. Let^(c) be they4A.-measure of the inverse image. ^ is a cylinder probability on H, and has characteristic fimction t(y) =/ f // 14 -Jy 15
Moments
Let F be a finite measure on H such that J H X H dF < Then J
or expectation of the measure F, and
X*, = Ex =J X dF. Definition 3.4. Let F be a Hilbert space valued basic
random variable. It has finite first moment if the following
conditions hold
(i) E > I < f, I < oo ^ for every 4 in H. (ii) E( < f, Definition 3.5. A basic random variable F has a finite second moment if (i) E ( < Ff (p > )^ , for every H. (ii) E ( Theorem 3.3. Let F be a basic random variable. If F has finite second moment it has finite first moment. Theorem 3.5. Let F = F-m where m=E(f). For any two elements X;y in H then E { < Ff < f,y )isa 16 bilinear form over H which is continuous. Hence there exists a bounded linear self-adjoint positive operator R such that, E ( ) = Example of Cylinder Set Measure. Let R be any self-adjoint nonnegative definite operator mapping H onto H. Define the measure“^on the Borel sets of any finite dimensional space M in the following way. Let e. ,e-,...e be an orthonormal basis in M. The 12 m Borel sets in M have a one-to-one correspondence with Borel sets in the "coordinate" space. That is X —> f^Xte^^.i = 1, 2, ...m| . Let denote the null space of R. Let M be any subspace of Nj^. Then the measure on any cylinder set with base in M is either 1 or 0 depending on whether it contains the zero element or not. Let be a complete orthonormal system in the range space of R. Let E^^ denote the cylinder set. En = 1 xj ,MX). Then since EnC/)(xi (< M^/ it follows that -K/ ^(e^)$[g(m/X^)J, where G(x) (exp-t^/2) dt and = min ‘^R^ ) 4^^ N is spanned by •Pi’-'-h- 17 18 Then —^ log^(E^) < log G(M/^). Hence liiaX log /a(E ) ^ log G (M//1 ) where A = lim if ?^0 then/*(E^)-^ 0. But if S(0,M) denotes the spheres with radius M and center at the origin then S(0,M) ^ for every n. This shows that there cannot he a countably additive measure on the class of Borel sets which coincides witly^on the cylinder sets. For if P denotes the countably additive measure then pCs(o,m)J^ P (E^). Since P(E^) = , P LS(0,M)J = 0 for every M. But H =^/S(0,'»^) and hence 1 = lim P 0^(0,^ )J which leads to a contradiction. Hence if R is any positive self-adjoint operator such that (^Rx,x;> ) 2. m ^x, x> , m ^o, then the induced cylinder measure cannot be countably additive on/^, assuming that H is not finite dimensional. It can be concluded that a necessary aind sufficient condition that the cylinder measure^h induced by R be countably additive is that for any complete orthonormal system (j>j^ Sup ^ ( < X, . To show this let the measure be countably additive. Then R 19 is nuclear that is n Conversely assume Sup J X ( < X, . But ^ H and hence Suggestions For Further Study. The concept of Probability Measures on a Hilbert Space has varied and extensive applications. An attempt has been made to study the basic properties in a very limited sense. However, a review of modern literature reveals that the concept can be extended to various areas which play vital roles in approximation theory and operational research. These concepts are briefly mentioned to suggest the possibilities of aspects which can be studied further. Linear Approximation Theory. Assiime that the random variables f,g have finite second moments and that there first moments are zero. Then Q(x,y) =E( where E denotes expectation, defines a continuous bilinear map. Hence there exists a bounded linear operator S suth that Q(x,y) = ^x, Sy > . Define the property (*) that the energy in a finite band goes 20 to zero. Then it can be verified that S* = E(fg*). If denotes the covariance of f, and Rg of g, then R^=E(ff*) and Rg=(gg*). Next assume that f is such that R^ is nuclear. Then it can be shown that S is Hilbert-Schmidt. Let L be a Hilbert-Schmidt operator mapping H into H. With R^ assvimed nuclear, then f - Lg is also a random variable, with finite second moment. It has first moment zero and its covariance is nuclear. Hence E( II f-Lg II ^)=TR(R^ +LRgL* - 2LS*)=Q(L). Note that Q(L) is a quadratic form in L in the Hilbert space of Hilbert-Schmidt operators. This can be viewed as the simplest linear estimation problem for Hilbert-space valued random variable. Another problem could be to define an inner product as an elementary random variable with nuclear covariance by Stochastic Process. A stochastic process in a Hilbert space is an indexed family of basic random variable. Let x(t w), t an element of the index set <^, be such a family with wt-ZL , an abstract space, the field and P the measure. Consider the class of all functions f(t) in a space Let S denote the smallest 21 field consisting of sets of the form ^xj x-feX, and x(t) -eF, for fixed t^ Where F is any Borel cylinder set in H. Consider now the mapping ^(w) = x(Qr, w) taiing-ZIinto X. Then the set ^~^(A) = |wj x(o(,w)-^:A, where A-^ s} belongs to Then P(A) -P(‘f"^(A) ) defines a finitely additive probability raeasxire on S. Then (X, S, p) is the stochastic. Stochastic Integrals. Let f be an element of LgCHiCO.T) ). Let W(t^w) be a Wiener Process with the following properties (i) E (W(t^,w) - W(t2 w) ) (W(t2,w)-W(t^,w) )* = 0 where (ii) For each t, W(t^w) is a continuousi linear transformation on H such that Then define the integral as follows for any function f(t)=c^ for t^ \'+i^x.= o 22 With the aid of the stochastic integral it can be shown that a Wiener Process with properties (i) and (ii) must have representation ^ ^ W(t,w) = j gj^i^d-it^ . BIBLIOGRAPHY Books Balakrishnan, A.V. Introduction to Q-ptimization Theory in a Hilbert Space. New Yorks Springer-Verlag,1971* Gelfand, I.M., and Vilenkin, N.Y. Applications of Harmonic Analysis. New Yorks Academic Press, 1964. Goldstein, A.A. Constructive Real .^alysis. New Yorks Harper and Row Book Company, 1967. Kelley, Namioka Linear Topological Spaces. New Yorks Van Nostrand, 1963• Lioms, J.L. Optimal Control of Systems Governed by tial Differential 'Eauations. New Yorks Diinod, 1969. Parthasarathy, K.R. Probability Measures on Ketric Spaces. New Yorks Academic Press, 1967• Riesz, R., and Nagy, S.Z. Functional Analysis. New Yorks Frederick-Ungar, 1965* Valentine, F.A. Convex Sets. New Yorks McGraw-Hill, 1964. Wilansky, A. Functional Analysis, New Yorks Blaisdell, 1964. Yoshida, K. Fi^ctional Analysis. New Yorks Springer- Verlag, I965. Young, L.C. Calculus of Variations and Optimal Control Theory. New Yorks N.B. Saunders Company, 1969* Articles Dunford, N. and Schwartz, J. "Linear Operators," Inter- science, 1 (1963)1 P* 48-59* Hille, E., and Phillips, R. "functional Analysis and Semigroups," AI