Integration, Measure and Probability

By H. R. Pitt

This text provides undergraduate mathematics students with an introduction to the modern theory of probability as well as the roots of the theory's mathematical ideas and techniques. Centered around the concept of measure and integration, the work is applicable to other branches of analysis and explores more specialized topics, including convergence theorems and random sequences and functions. 1963 edition.

• Language: English
• Publisher: Dover Publications
• Release date: May 13, 2013
• ISBN: 9780486282527

Book preview

INDEX

Chapter 1

SETS AND SET-FUNCTIONS

§ 1. SET-OPERATIONS

The modern theories of measure and probability are based on fundamental notions of sets and set-functions which are now familiar and readily accessible. It is therefore simply on grounds of convenience that we begin with a brief summary of these ideas.

of elements (points) x, subsets X, X1 X, and certain systems of subsets. In accordance with common usage, we write x X to mean that the point x belongs to the set X and XX2 to mean that the set X1 is included in (or is a subset of) the set X2. The empty set is denoted by 0, and it is plain that X X X . Moreover, XX2 and XX3 together imply that XX3, while XX2 and XX1 imply that X1 = X2.

The operations which are not in X is called the complement of X and written X′. It is clear that (X′)′ = X′ = 0 and, if XX2, then XX′1.

The union of any collection of sets is the set of points x which belong to at least one of them. The collection need not be finite or even countable. The union of two sets X1, X2 is written XX2. The union of a finite or countable collection of sets Xn (n . It is plain from the definition that the union operation does not depend on any particular ordering of the component sets and that there is no limit process involved even when the number of terms is infinite.

The intersection of a collection of sets is the set of points which belong to every set of the collection. The intersection of two sets is written XX2, while the intersection of a sequence Xn . A pair of sets is disjoint if their intersection is 0 and a collection of sets is disjoint if every pair is disjoint. In particular, the union of a sequence of sets X1, X

The difference X1 − X2 = XX2 between the sets X1 and X2 is the set of points of X1 which do not belong to X2.

It is clear that each of the union and intersection operations is commutative and associative, while each is distributive with respect to the other in the sense that X (XX2) = (X X(X X2), X (XX2) = (X X(X X2) or, more generally,

. Furthermore, the operations are related to complementation by

and, more generally,

The sequence of sets Xn is increasing (and we write Xn ) if, for each n, Xn Xn+1, and decreasing (Xn ) if XnXn. The upper limit, lim sup Xn, of a sequence is the set of points belonging to infinitely many of the sets; the lower limit, lim inf Xn, is the set of points belonging to Xn for all but a finite number of values of n. It follows that lim inf Xn lim sup Xn. If lim sup Xn = lim inf Xn = X, X is called the limit of Xn and Xn is said to converge to X. We then write Xn → X, or Xn X, Xn X in the cases when Xn . In particular, if Xn decreases, then

Similarly, if Xn .

§ 2. ADDITIVE SYSTEMS OF SETS

A non-empty collection S of sets X is called a ring if, whenever X1 and X2 belong to S, so do XX2 and X1 − X2. The empty set can be expressed as 0 = X − X and therefore belongs to every ring. Moreover, since XX2 = (XX2) − {(X1 − X(X2 − X1)}, the intersection of two sets of S also belongs to S. The result of applying a finite number of union, intersection or difference operations to elements of a ring is therefore to give another element of the ring. In other words, the ring is closed under these operations.

itself belongs to the ring, but we do not assume that this is the case in general.

A ring S which and has the further property that the union of any countably infinite collection of its members also belongs to S is -ring. Since

-ring also contains the intersection of any countable collection of its members and is thus closed under union, intersection, difference and complement operations repeated a finite or countably infinite number of times.

-ring to include a given collection T of sets of X and the following theorem is fundamental.

THEOREM 1. A given collection T of subsets of space is contained in a unique minimal ring -ring) which is contained in every ring -ring) which contains T.

The minimal ring is called the ring generated by T. -ring generated by T is called the Borel extension of T, and its members are called Borel sets.

There is at least one ring containing T, -ring for ring throughout. In applications, T is -ring. The theorem still holds in a trivial sense if T is -ring, but it is then its own Borel extension.

of real numbers, the intervals do not form a ring since the union of two intervals need not be an interval. However, the collection of sets which consist of a finite union of intervals of type a x < b k of real vectors (x1, x2, …, xk) in which a finite union of rectangles aj x < bj (j = 1, 2, …, k) is called a figure and the ring generated by the rectangles is the ring of figures.

The ring of figures, particularly in the case n = 1, is of fundamental importance in the theory of measure and integration. The ring is easy to visualise and provides useful concrete example on which the significance of abstract theorems may be illustrated.

§ 3. ADDITIVE SET FUNCTIONS

(X) is a function whose range of definition is a system (usually a ring) of sets X according to the following scheme.

are not defined.

(X) defined in a ring S is called additive if

for every finite disjoint collection of sets in S. The function is completely additive in S if the additivity property holds also for countably infinite collections of sets in S provided belongs to S. This last proviso is not needed when S is -ring since it is satisfied automatically. Otherwise, it must be retained and only certain sequences of sets (those whose union belongs to S) may be admitted.

(X) is finite for at least one set X(X(X (Xis called finite.

(X-ring is called a measure. It is called a probability measure ) = 1.

is said to be continuous from below at X (Xn(X) whenever Xn X. It is continuous from above at X (Xn(X) whenever Xn X (XNfor some N. It is continuous at X if it is continuous from above and below at X unless X = 0, in which case continuity means continuity from above. The relationship between additivity and complete additivity is expressed in terms of continuity in the following theorem.

THEOREM 2. A completely additive function is continuous. Conversely, an additive function is completely additive if it is continuous from below at every set or if it is finite and continuous at 0.

-ring.)

is completely additive. If Xn X (Xn