Probability and Mathematical Statistics
A Series of Monographs and
Textbooks
Edited by
Z. W. Birnbaum
E. Lukacs
University of Washington
Seattle, Washington
Catholic
University
Washington, D.C.
1. Thomas Ferguson. Mathematical Statistics: A Decision Theoretic
Approach . 1967
2. Howard Tucker. A Graduate Course in Probability. 1967
3. K. R. Parthasarathy. Probability Measures on Metric Spaces. 1967
4. P. Révész. The Laws of Large Numbers. 1968
5. H. P. McKean, Jr. Stochastic Integrals. 1969
6. B. V. Gnedenko, Yu. K. Belyayev, and A. D. Solovyev. Mathematical
Methods of Reliability Theory. 1969
7. Demetrios A. Kappos. Probability Algebras and Stochastic Spaces.
1970
PROBABILITY ALGEBRAS
AND
STOCHASTIC SPACES
Demetrios A. Kappos
UNIVERSITY OF ATHENS
ATHENS, GREECE
1969
ACADEMIC PRESS
New York and London
COPYRIGHT © 1969, BY ACADEMIC PRESS INC.
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RIGHTS RESERVED.
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STANDARD BOOK NUMBER: 12-397650-2
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AMS
1968 SUBJECT CLASSIFICATIONS 2865, 4606, 4610, 6005
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
During the last twenty years, many excellent textbooks have enriched
the mathematical literature on probability theory. With the help of
classical set-theoretic measure and integral theory, these books introduce
the fundamental concepts of probability theory, then formulate and study
old and recent problems germane to the theory. There is however an
alternate way to introduce the main notions of probability theory, a way
which is more naturally adapted to the empirical origins of the subject.
The present volume is an exploration of this alternate development.
There are three fundamental notions of probability theory: Event,
probability of an event, random variable. A given set of events forms a
Boolean algebra with respect to the logical connectives (considered as
operations) " or ", " and ", and " not ". Probability is a normed measure
on a Boolean algebra of events. It is natural to consider probability
as finitely additive and strictly positive, i.e., equal to zero only for the
impossible event. Therefore, a Boolean algebra 51 endowed with a
finitely additive and strictly positive probability p can be considered as a
probability algebra (Si,/?). In all empirical cases a Boolean algebra of
events can be endowed with an additive and strictly positive probability.
Moreover, the accountable) additivity of probability, which has important
mathematical consequences in the theory, can always be obtained by a
metric extension of a probability algebra (SI,/?) to a probability σalgebra (SÏ, p), in which SÎ is a Boolean σ-algebra and p a countably
additive and strictly positive probability.
It is well-known that a Boolean algebra St is always isomorphic to a
field (Boolean algebra) of subsets of a set ( = space) Ω. Thus the investiv
VI
PREFACE
gation of events and their probabilities can be reduced to a study of
normed measures on fields of sets. Moreover, it is always possible to
represent the Boolean algebra 31 of events by a field A of subsets of a set,
so that the normed measure (the probability) P on A is set-theoretically
countably additive. Let (Ω, A, P) be a so-called probability space,
which represents, set-theoretically, a probability algebra (31,/?); then
(Ω, A, P) can always be extended to a complete probability σ-space
(Ω, Ä, P), in which A is a σ-field containing A and P a complete, normed,
countably additive measure on Ä. The elements of Ä can be considered
as events; in the case, however, in which the cardinality of Ω is > K0 ^
may happen that there exist non-empty sets EeA (i.e., events different
from the impossible event) of measure ( = probability) zero, which have
no probabilistic interpretation. We can overcome this difficulty by
considering directly the probability σ-algebra (3Ï, p) instead of the probability σ-space (Ω, Ä, P).
Our aim is to develop the fundamental notions of probability theory
in this " point-free " way. This, however, requires knowledge of lattice
theory. We find that lattice theory also provides simplicity and generality,
since it deals with classes of random variables which are the elements of
the so-called stochastic spaces. The space of all elementary random
variables defined over a probability algebra in a " point-free " way is a
base for the stochastic space of all random variables, which can be
obtained from it by lattice-theoretic extension processes. There are,
however, problems in which one wants to consider individual samples
and cannot work without points; then one can always assign a suitable
probability σ-space to the probability algebra under consideration. Conversely, one can assign to every probability σ-space (Ω, A, P) a probability
σ-algebra (3ί, p) in which 31 is the quotient Boolean σ-algebra A/N,
where N is the σ-ideal of all sets of probability zero. Thus the two
theories are equivalent.
In the lattice-theoretic treatment of probability theory, a structural
classification of all possible probability algebras (therefore an analogous
classification of all possible stochastic spaces) can easily be obtained,
which provides us with a representation of every probability σ-algebra
by a probability σ-space (Ω, A, P) in which Ω is the cartesian product of
factors equal to the interval [0, 1] of the real line, A a σ-subfield of the
σ-field of all Lebesgue product measurable subsets of Ω, and P the product
measure of A. A corresponding representation of random variables by
Lebesgue measurable functions defined on Ω can also be obtained.
PREFACE
vu
The origin of this book is in a set of lectures which I gave in the academic
year 1963-64 at the Catholic University of America, Washington, D.C.
The Statistical Laboratory there issued a mimeographed version of my
notes under the title " Lattices and their Applications to Probability ".
The present text is a revised and expanded version of these notes, maintaining the central mathematical ideas of the lectures, namely probability
algebras and stochastic spaces (i.e., spaces of random variables). The
part of the notes devoted to pure lattice theory has been shortened and
the most important concepts and theorems of this theory have been
stated in two appendices, mostly without proofs.
In addition to the material in the mimeographed edition, the present
volume contains a general way to introduce the concept of random
variables taking values in spaces endowed with any algebraic or topological structure. In particular, we study the cases in which the space of
the values is a lattice group, or vector lattice. When the space of the
values is a Banach space, a theory of expectation and moments is stated.
A theory of expectation can be easily stated in more general cases of
spaces of values: for example, locally convex vector spaces or topological
vector spaces, for which an integration theory is known.
We have restricted ourselves on the introduction and study only of the
fundamental mathematical notions. We mention only a few facts about
the concepts of independence and conditional probabilities and expectations. Certainly, it would be interesting to state the theory of stochastic
processes and, especially, the theory of martingales. But this would go
beyond the scope of the present monograph, or it would have to be
published in a second volume.
The author wishes to express his gratitude to Dr. Eugene Lukacs of
the Catholic University, who made it possible for him to give lectures
and publish them. It is a pleasure to offer thanks to F. Papangelou and
G. Anderson who read critically the manuscript of the lectures and made
valuable suggestions during the mimeographed edition of them.
He expresses his best thanks to Miss Susan Papadopoulou who read
all the manuscript of the present edition carefully and made valuable
suggestions. In many cases she has helped him to give simpler formulations and proofs.
Thanks are finally due to Mr. Constantine Halatsis for his efficient
typing of the manuscript and the composition of the several indices.
Athens, Greece.
April 1969.
D. A. Kappos.
I
PROBABILITY ALGEBRAS
1. DEFINITIONS AND PROPERTIES
1.1. A probability algebra (pr algebra) (St,/?) consists of a nonempty
set St of elements denoted by lower-case latin letters: a, b, c,
...,x,y,...,
called events and a real-valued function p on St, called a probability (pr).
In the set St two binary operations avb (a orb) and aAb (aandb) and
one unitary operation ac (not a) are defined, which introduce in St the
algebraic structure of a Boolean algebra.t
The probability p satisfies the following conditions:
1.1.1. p is strictly positive, i.e., p(x) ^ 0, for every x e S l and p(x) = 0
if and only if x = 0 , where 0 is the zero of 91.
1.1.2. /? is normed, i.e., />(e) = 1, where e is the unit of SI.
1.1.3. /? is additive i.e., /?(a v 6) = /?(a) +/?(&) if a
anc
* * a r e disjoint.J
We shall call the unit e the sure event and the zero 0 the impossible event
of the event algebra St. Every x e St with x ^ 0 and x Φ e is called a
possible event of the algebra St.
Î We consider the theory of Boolean algebras as known ; cf. also Appendix 1 of this
book.
{ We say that the event a and the event b are disjoint (exclude each other, or are mutually
exclusive, or are incompatible) if a Ab = Ö.
1
2
1.2.
I. PROBABILITY ALGEBRAS
Properties of the prp.
1.2.1. If x l 5 x 2 , ...,x n are pairwise
disjoint
events
of
9ί,
then
p(xt v x 2 v ... v xn) = pixj +p{x2) + . . . +p(xn).
Proof by induction.
1.2.2. If x < y, then/?(x) ^ p(y) and, moreover, p(x) < p(y) if x Φ y
(i.e., if x < y, then /?(x) < p(y)).
Proof. If x < y, then z = y — x is such that y = x v z and x and z are
disjoint. According to 1.1.3, we have p{y) = /?(x)+/?(z), i.e., /?(x) < p(y).
If x # j , then z ^ 0 and p{z) > 0, i.e., /?(x) < p(y).
1.2.3. If x < y, then /?0 - x) = p(y) -/?(x). In fact, according to the proof
of 1.2.2: p(y)=p(x)+p{z\
i.e., p(z) =p(y-x)
=p{y)-p{x).
1.2.4.
For every pair x e 2t and j> e 21, we have
p(x) +p(y) = p(x v y) +/>(x Λ J ) .
In fact, we have
xv y = (x—x/\y)y{y — x/\y)v
{x/\y).
The terms of the join on the right are pairwise disjoint, thus
p(xvy)
i.e.,
1.2.5.
=
p(x-xAy)+p(y-XAy)+p(xAy)
= p(x)
-P(X
Λ y)
+p(y) -ρ(χ
=
p(x)+p(y)-p(xAy);
p(x) +p(y) = p(x v y) +p{x
A
A y)
+p(x
A
y)
y).
For every x e 21 we have p(xc) = 1 — p{x). Proof obvious.
1.2.6. We have p(x) = 1 if and only if x = e. Proof obvious.
By induction, one can prove the following generalization of property
1.2.4:
1.2.7. For three, or more generally for n ^ 2 events xv e 3i, v = 1, 2,
...,« we have:
Σ/>(*) = £/>( V Λ*„).
v=l
v=l
\pes«.vi<v
/
η 2
>
2. PROBABILITY SUBALGEBRAS
where Sn,v is the set of all finite sequences p = {p l5 p 2 , ··· > Pv}
1 < Pi < Pi < ··· <Pv < " ·
3
w tn
^
2. PROBABILITY SUBALGEBRAS
2.1. Let (31,/?) be a pr algebra and S3 a Boolean subalgebra of 31; then
the restriction of the function p to S is a probability on 33. The pr algebra
(33,/?) is then called a probability subalgebra of (31,/?). Hence to every
Boolean subalgebra © of 31 there corresponds a pr subalgebra (33,/?) of
the pr algebra (31,/?).
2.2. Let 8 be a non-empty subset of 31; then there exists a smallest
Boolean subalgebra, b (S), of 31 containing S ; the pr subalgebra (b (S),/?)
is then called the pr subalgebra generated by S in (31,/?).
2.3. There is a pr subalgebra of (31,/?), namely, the pr algebra (U,/?)
where U = {0, e) and /?(0) = 0, p(e) = 1. We call (U,/?) the improper
pr algebra.
2.4. Remark, Throughout this book every pr algebra and every pr subalgebra will be supposed as not improper.
2.5. Every possible event x e 3 I , i.e., x φ 0 , x Φ e, generates a pr
subalgebra ( ϊ , /?) in (3ί, /?), where X = {0, x, xc, e).
3.
ISOMETRIC PROBABILITY ALGEBRAS
3.1. Let (SIi,/?i) and (3I2,/?2) be two pr algebras. We say that the pr
algebra (φ^Ρι) is isometric to the pr algebra (3I2,/?2) if and only if
there exists an isomorphic map φ of the Boolean algebra 31A onto the
Boolean algebra 3I2 such that Ρι(χ) = Ρι(Φ(χ)) f° r every ΧΕ^&^
4.
EXAMPLES
4.1. Let E = {al9 a 2 , ..., an} be a finite set of points au a2, ..., an, n ^ 2.
We take the class ^3(£) of all the subsets of E to be the Boolean algebra
<$\n = S$(E). Let/? l5 /? 2 , ...,/?„ be positive real numbers with 0 </? v < 1,
v = 1, 2, ..., /7, and ρί+ρ2 + ~>+Pn = 1· F ° r every subset
4
I. PROBABILITY ALGEBRAS
of E9 we define the probability as follows:
/>({*/,» <*i2> · · ·> tf/J) = Ph +Ph + ··· + A v
and we define, for the empty set, p(0) = 0. Then (3In,/?) is a pr algebra
with a finite number (2n) of events.
4.2. Let E = {al9 al9...,} be a denumerable set of points av9 v = 1,2,....
Let, moreover, pv9 v = 1, 2, ..., be positive real numbers with 0 < pv < 1,
00
v = 1,2,..., and £ /?v = 1.
v=l
We take the class φ ( £ ) of all subsets of E to be the Boolean algebra 2IXo
and assign to every finite subset \ah9 ai2, ...,aik} £ E or infinite subset
{Ö/Ptfy2,...,} £ £ the probability defined by
or
p({aii9ai29 .~,aik})=pil+pi2
+ ...+pik
v= l
respectively. We define, for the empty set, p(0) = 0.
Then (Αχο9ρ) is a pr algebra with an uncountable number of events
because, as it is well known, the cardinality of A#0 = φ(Ε), with E a
denumerable set, is equal to the cardinality of the continuum, i.e., > K0.
4.3. Let X = {Xi}, iel, be a class of symbols xi9 iel9 where / is a set of
indices with a cardinality |/| = K ^ K0 ; then there exists a Boolean algebra
©a, the so-called free Boolean algebra S K with X generators, which is
characterized by the following properties :
(a) The symbols {JCJ, i e J, are elements of ©N and the class X = { x j ,
/ e J, generates the Boolean algebra 93x.
(β) Every map φ of the class X into an arbitrary Boolean algebra
© can always be extended to a homomorphism of 33x into 23.
We shall not prove this statement, but we shall mention some facts
about the algebraic structure of 33K that are important for the definition
of a probability on 93^.f
Let {il9 il9 ..., /„}, n ^ 1, be a finite subset of/; then we call the expression y = yiiAyhA...Ayin9
where yiv = xiv or xciv9 v = 1, 2, ..., n9 a
t Details about free Boolean algebras one can find in Sikorski [5], §14.