Integration, Measure and Probability

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.

We shall be concerned with a space′ of elements (points) x, subsets X, X1 X2, … of , 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 X1 X2 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 and 0 X . Moreover, X1 X2 and X2 X3 together imply that X1 X3, while X1 X2 and X2 X1 imply that X1 = X2.

The operations on sets are defined as follows. The set of points of which are not in X is called the complement of X and written X′. It is clear that (X′)′ = X, 0′ = , ′ = 0 and, if X1 X2, then X′2 X′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 X1 X2. The union of a finite or countable collection of sets Xn (n = 1, 2, …) is written . 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 X1 X2, while the intersection of a sequence Xn is written . 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, X2, … can be expressed as the union of the disjoint sets

The difference X1 − X2 = X1 X2 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 (X1 X2) = (X X1) (X X2), X (X1 X2) = (X X1) (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 Xn+1 Xn. 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 decreases or increases. It is plain from the definitions that lim inf , lim sup . In particular, if Xn decreases, then

Similarly, if Xn increases, .

§ 2. ADDITIVE SYSTEMS OF SETS

A non-empty collection S of sets X in is called a ring if, whenever X1 and X2 belong to S, so do X1 X2 and X1 − X2. The empty set can be expressed as 0 = X − X and therefore belongs to every ring. Moreover, since X1 X2 = (X1 X2) − {(X1 − X2) (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.

The ring is also closed under the complement operation if (and only if) itself belongs to the ring, but we do not assume that this is the case in general.

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

it follows that a -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.

We shall often be concerned with the construction of a ring or -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. The minimal -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, namely the ring of all subsets of . We consider all such rings. It is plain that their intersection, consisting of the sets which belong to every such ring is itself a ring and has the properties stated, and the proof remains valid if we substitute -ring for ring throughout. In applications, T is usually a ring but not a -ring. The theorem still holds in a trivial sense if T is itself a -ring, but it is then its own Borel extension.

If is the space 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 do form a ring which is plainly the ring generated by these intervals in the sense of the last theorem. This ring does not contain and is not a -ring. There is a straightforward generalisation to the space 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

A set function (X) is a function whose range of definition is a system (usually a ring) of sets X and whose values belong to some appropriate space. We shall deal in this book only with set functions whose values are real numbers, but it is convenient to augment these by appending the elements ± and giving them algebraic and order properties in relation to real numbers according to the following scheme.

The operations − , / , 0. are not defined.

A set function (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 also that belongs to S. This last proviso is not needed when S is a -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.

We shall always assume that (X) is finite for at least one set X, so that (X) = (X 0) = (X) + (0) and (0) = 0. It is not possible for an additive set function to take the value + on one set and − on another, since its value on their union would then be undefined; and we shall exclude this possibility by admitting + as a possible value, but not − . A set function which takes neither of the values ± is called finite.

A non-negative and completely additive function (X) defined over a -ring is called a measure. It is called a probability measure if ( ) = 1.

A set function is said to be continuous from below at X if (Xn) → (X) whenever Xn X. It is continuous from above at X if (Xn) → (X) whenever Xn X and (XN) < for 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.

(The ring in which the function is defined may, but need not, be a -ring.)

Suppose that is completely additive. If Xn X and (Xn) ≠ ± for