Measure and Measurable Assemblages. —This subject can best be considered by limiting our discussion to sets of numbers in the interval between 0 and 1. Let P be such a set, and let Q be the complementary set — the set of all numbers between 0 and 1 that are not in P. Suppose that a denumerable number of inter vals — i.e., sets of all the numbers between k and k +e, exclusive of these values — have been found, including all the points of P. These intervals have lengths, and these lengths, added in the descending order of magnitude, form a series. If this series is convergent, it has a sum, which we shall call S; if the series diverges, we shall say that S is infinite. S de pends in general on the particular choice of in tervals containing P. However, S will have either a minimum value or a lower limiting value. This is called m (P), or the upper measure of P, and depends on P alone. If 1 — (Q) — the lower measure of P —is equal to m (P), their common value is called m (P), the measure of P. A set which has a measure is said to be measurable. The theory of meas urable sets is of the utmost importance in the modern discussion of integration (see REAL VARIABLE, THEORY OF FUNCTIONS or). Whether there are sets that are not measurable is un known; an apparent exemplification of such sets by Van Vleck rests on the precarious axiom of Zermelo. Every limited closed assemblage is measureable.
Improper Infinite and Proper Infinite, or Transfinite.— The ordinary notion of mathe matical infinity is that of a finite variable, such as tan a, which can take a finite value greater than any previously specified finite value; and such an equation as tan 90°= co is understood by mathematicians to be a kind of short-hand for saying that, by taking a near enough to 90° tan a can be made to exceed any preassigned finite number, and it does not mean that co is a value that tan a may assume. Similar illus trations abound. Such a variable as thus remains always finite but may be made large at will is sometimes described as an infinite '(variable) in analogy with the reciprocal notion of infinitesimal, a variable that remains always finite but may be made small at will. Such infinites are described by Cantor as improper infinites. On the other hand both geometry and analysis have long recognized another sort of infinite, viz., one that is not variable but is constant. Such an infinite, for example, is the distance from any finite point of a range (see PROJECTIVE GEOMETRY) to the point common to the range and any parallel range. Another example is the distance from an finite point of the complex plane to an n 'tte points of the plane (see COMPLEX VARIABLE). Such in finites are proper infinite:. Proper infinites of
a very different sort arise in the theory of as semblages. We shall now give an account of their genesis and nature.
Transfinite Cardinal Numbers and their Laws.— Denote by A any assemblage of ele ments a; symbolically, On disregard ing both the character of the a's and any and every order of their arrangement, a new as semblage, an orderless assemblage of character less elements (units), arises, called the power or cardinal number of A and denoted by the symbol A.* Herewith the term power (Mach tigkeit) is itself defined; sameness of power was defined above. Plainly every class has a definite power, or cardinal number. The equation =-_ = A = B means that A and B have the same or equal powers or cardinal numbers. It is easier seen that, when and only when B, A=B. If A, B, C, . . . have no common element, the assemblage of all the elements involved will be denoted by (A, B, C, . . .). If also A', B', C', . . . have no common element, and if A..- A', B B', C . . . , then (A, B, C, . . (A', B', C', . . .), and the cardinal numbers of these composite assemblages are equal, or the same.
Notion of Greater and Less Powers or Cardinals.— If A and B are such that A has no part equivalent to B and that B has a part equivalent to A, the cardinal number of A is said to be less than that of B, that of B greater = — = than that of A; symbolically, A < B, or B >A.
are cardinal numbers, and if a < /3, and P < y, then a < y. Any one of the relations a=13, a < /3, a > /3, excludes the other two. But it does not follow that every pair of cardinals a and # must satisfy one of the three relations, and it is not known whether they satisfy one of them. This last proposition belongs to the theory of assemblages, a term explained at a later stage of this writing.
Addition of Powers or Cardinals.— If a and p be the cardinal numbers of A and B, A and B having no common element, and if y be the power of (A, B), then Such is the definition of addition. As a power in an orderless assemblage,a + P=P and, in case of any three powers, a + + = (a that is, addition of powers is commutative and associative.
Multiplication..—Let A=.3 a 1 and /1=-1 DI, Associate each a with each b. Consider each pair (a, b) as an element. The assemblage of these is denoted by (A .B). Hence (A .B)= 1(a,b)1. The power y of this last obviously depends only on the powers a and b of A and B. Hence the definition of product: a•P--y. As the power, or cardinal number, of an assem blage is orderless, it is readily seen that a and that, for any three powers, a • (a-11)-y, a• + + y; that is, multiplication of powers is commutative, associative and dis tributive.