POINT SETS. A point set is a collection of points selected from a given space. The study of the properties of point sets constitutes that branch of mathematics known as point sets, or the theory of sets of points. Generally speaking, the properties of a point set may be classified under two heads, (I) topological and (2) metric. For a description of the former see ANALYSIS SITUS. A brief introduction to the metric properties of point sets is given below.
In order to approach the subject by as simple an example as possible, let us confine ourselves to the case where the given space is an ordinary straight line, L. If P and Q are distinct points of L, then the point set consisting of P and Q together with all points between them is called an inter val and is denoted by [P,Q]. Let us imagine that we have a com mon foot-rule which can be applied to L in order to measure lengths. Then given an interval [P,Q] we can measure its length, and say that it is a certain number of feet. Of a single point we would say, in accordance with the ordinary geometry notion, that its length is zero. If we are given two intervals which have no point in common, it is not natural to speak of the length of the set of points which they represent, the word "length" being usually applied only to connected pieces. In this case we shall use the word "measure," and say that the measure of this point set is the sum of the lengths of the two intervals.
However, when we speak of a point set on L, this does not necessarily imply that we are thinking of an interval, a single point, or a set of intervals ; we sometimes mean to indicate a set of points which contains no connected portion, i.e., which contains no interval. One might be tempted to say that since a point has length zero, the "measure" of such a set would be the sum of the lengths of its individual points, i.e., the sum of a set of zeros, and hence zero. Such a hasty decision would not lead to very fruitful results, however, for the following reason. If we determine upon a "measure" for two point sets, A and B, which have no points in common, the sum of their measures should naturally be the meas ure of the point set which is made up by them taken together. Thus, above, we have stated that the measure of a set consisting of two intervals with no common point is the sum of the lengths of those intervals. Now any interval [P,Q] can easily be shown to be the sum of two sets A and B each of which fails to contain any interval, and if we arbitrarily call the measure of both A and B zero, the sum o' their measures would be zero, which is not the length of [P ,Q], no matter how small the length of [P ,Q]. In other words, we want a measure of a set of points which will cor respond to the ordinary idea of length.
We have now introduced what is known, in the theory of sets of points, as the problem of measure. There have been several methods devised for finding a measure of an arbitrary set of points. We shall describe, briefly, the theory of Lebesgue meas
ure, which is the foundation of the Lebesgue theory of integration.
A set, A, is said to be covered by a col lection, G, of intervals, when every point of A is in some interval of G. If the set of intervals G is denumerable, then we shall say that it is a covering of A. (A set is called denumerable if its ele ments can be "tagged" with positive integers in such a way that no two elements of the set are "tagged" with the same integer.) If the sum of the lengths of the intervals of G exists, let us call this the sum-length of the covering. Now of all possible coverings of A consider the corresponding sum-lengths, and let N be the largest number which is not greater than any of these sum-lengths. Then N is called the exterior measure of A and is denoted by
Suppose, now, that [P,Q] is some interval, whose length we shall denote by d, such that all points of A are within [P,Q]. Let B be the set of all points of [P,Q] that do not belong to A, and let
denote the exterior measure of B, found just as
was found. If it happens that
then
is accepted as the measure of A, and is what is known as the Lebesgue measure of A. Of course we have at the same time that
is the Le besgue measure of B, and in accordance with our ideas of length we have required that the sum of the two measures give the length of [P,Q]. To be sure, the Lebesgue measure of a set of points may not exist, but it does exist for all ordinary point sets. Indeed it is not at all easy to give an example of a set of points which has no Lebesgue measure, and all of those examples which have been given make use of certain methods which are held to be unacceptable by many mathematicians.
For the measure of a set of points in a plane, areas are em ployed. Thus, the measure of the set of all points in a square is the area of the square. And to get the measure of a general plane point set M, a covering of M is made by means of squares. In three dimensions cubes are employed, and we deal with sum volumes.
The introduction of the notion of measure has led to an enrich. ing of the content of general analysis that could hardly have been realized otherwise. And the effect has been felt not only in mathe matics itself, but in the closely allied fields of mechanics and dynamics.
der mathematischen Wissenschaften, Band 113, Heft 7 (1924). In ss. 18-2oc will be found a summary of various kinds of measure. On p. 858 are given references to applica tions of the notion of measure to problems in physics, astronomy, etc. E. W. Hobson, The Theory of Functions of a Real Variable (2nd ed. 1921) vol. i., p. 158 seq.; A. N. Whitehead, An Introduction to Mathe matics (19"), a popular treatise, on pp. 77-79 of which will be found an example of a set which is denumerable, as well as of a set which is not denumerable; W. H. and G. C. Young, The Theory of Sets of Points (1906). (R. L. WO