technical a character for rapid summary. The* real theory, on the other hand, has been remodelled from its foundations. The older form of the theory was cumbrous and unattractive. The modern theory has the aesthetic character required of a first- rate mathematical science, and its development has been per- haps the most striking achievement of modern analysis.
1. Sets of Points. The theory of functions of a real variable is based upon the theory of aggregates (see 19.847-850) and in particu- lar the theory of " sets of points." A set of points f is an aggregate of real numbers DC, such as the aggregate of rational numbers, or of irrational numbers, in the interval (o, l). A number is said to be a " limit point " (Hdufungstelle) of f if every " neighbourhood " of , that is to say every interval ( e, +e) including , contains points of f other than itself. A limit point of f may or may not belong itself to f. Thus-every number of (o, l), rational or irrational, is a limit point of the set f of rationals of (p, l). If every limit point of f belongs to f, f is closed. If every point of f is a limit point, f is compact or dense. A set which is both closed and compact is perfect. In particular, the continuum, the aggregate of all real numbers, is perfect: this is the first and most striking stage in G. Cantor's mathematical analysis of the continuum.
An idea of dominating importance in the theory of functions is that of the content or measure of a set of points. Suppose, for sim- plicity, that the set f in question is contained in (o, l). Then Cantor defined the content of fas follows: Divide (o, i) in any manner into a finite number of intervals 5, and these intervals d into two classes S t and 8 2 , according as they do or do not include points of f ; and let c(&) be the sum of the lengths of the intervals 81. Then the content of f is the limit of c(5) when the intervals 5 tend to zero, if this limit should exist.
There is a striking defect in this definition, the full implications of which were first perceived by E. Borel. The content of the sum of two sets is not generally the sum of their contents. Thus the rationals of (o, i) have content I (since every 8 is obviously a Si), and likewise the irrationals. The sum of the contents is 2, whereas the content of the sum is I. The rationals of (0,1) cannot be included in a finite set of intervals whose aggregate length is less than I. If we abandon the restriction that the set of intervals must be finite, the situation is completely changed. Thus Borel observed that we may include the
rational p/q in the interval ( -4, +Ji) > an d that the sum of all
these intervals may be made as small as we please by choice of ; and this simple remark has revolutionized the theory of functions. The first step was to frame a satisfactory definition of measure, and this concept, which has entirely superseded Cantor's " content," is now defined as follows. We consider sets f included in (o, i). Let f be enclosed, in any manner whatsoever, in a system of of intervals 4; let m(a) be the sum of the intervals of CT; and let m, be the lower bound (or " inferior limit ") of the aggregate of values of m( Then m e is the exterior measure of f. The interior measure mi is I m',, where m' e is the exterior measure of f , the set complemen- tary to f, i.e. the set of points of (0,1) which do not belong to f. If m e = nti, the set f is measurable, and its measure is m, the common value of m, and mi. This definition (due to H. Lebesgue) is of ex- treme generality, and no example of a non-measurable set is known. Measure, thus defined, has the properties which measure ought to have, but which Cantor's content lacked. In particular the sum of two mutually exclusive and measurable sets is measurable, and its measure is the sum of the measures of the component sets. The measure of any enumerable set, and in particular of the rationals, is zero. The definition may be extended to sets in space of any num- ber of dimensions.
2. Integration. The new theory of measure has led to new theories of integration, in the light of which the older theories are of historical or didactic interest only. The most important of these theories are due to H. Lebesgue and W. H. Young.
(o) Lebesgue's definition of an integral is as follows. A function f(x), defined in an interval (a,b), is measurable if the set of points S (A) for which/> A ismeasurable for every A. All known functions are measurable. We now suppose that / is bounded, so that (say) h and we divide up the interval (h, H) into a finite number of intervals (/,-, U+i or 81. It is this subdivision of the range of variation of /(*). instead of (as in the older theory) that of x, that is charac- teristic of Lebesgue's procedure. The set of points for which (/(^ f is measurable. If we denote its measure by ,-, write J = 2/,-m,- and suppose that the intervals 81 tend to zero, then J tends to a limit I, and we write:
1 =
The integral so defined is a bona-fide generalization of the integral of Riemann, for it exists whenever Riemann's integral exists and agrees with it in value. But it is far more general : thus the function f(x) which is unity when x is a rational of (0,1), and zero otherwise, has no. Riemann integral, but has a Lebesgue integral equal to zero. The definition is capable of many-sided generalization, to unbounded functions, and functions of many variables; it throws entirely new
light on the relations between integration and differentiation; and it has proved itself adapted for a mass of analytical applications of the most far-reaching importance, in particular in the theory of Fourier's series and the theory of integral equations.
(b) A different definition was proposed by W. H. Young. He ad- heres to a subdivision of the range of variation (a, b) of the independ- ent variable; but, instead of dividing it into a finite number of inter- vals, divides it into a finite or infinite number of measurable sets. This procedure leads to results roughly equivalent to those of Lebesgue's theory; but it is somewhat more general; and is cer- tainly a more natural development of the older theory of measure.
3. Geometrical Applications. Those new theories have led in- evitably to a searching reexamination of the concepts of " curve," " surface," " length," " area," and so forth, which were generally accepted without question by the older analysts on the supposed evidence of geometrical intuition. This unreflective attitude has now been abandoned, and it is recognized that analysis is in no sense dependent upon geometry. The notion of a curve was first made precise by C. Jordan. A curve is a set of points (x,y), that is an aggregate of pairs of real numbers x, y where x and y are functions of a single variable /, subject to appropriate restrictions. A simple closed continuous curve is a curve for which (i) x = x(t) and y = y(t) are continuous for /i^i^fe, (2) x(h)=x(t\) and y(h) =y(h) and (3) it is false that x(t') =x(t") and y(t) = y(/") for any pair of values t', t" other than h, h. 'A fundamental theorem, duein substance to Jor- dan, asserts that such a curve C divides the plane into two " regions " D and D' separated by the curve. Two points which lie in the same region can be connected by a continuous curve which has no point in common with C; but points which lie in different regions cannot be thus connected. We thus define the inside and outside of a closed curve in strictly analytical terms. A similar account has been given of the concepts of area and length. In particular the simple closed continuous curve C has both an area and a length if x(t) and y(t) are functions of bounded (or limited) variation.
AUTHORITIES. More or less complete accounts of the modern theories will be found in : E. W. Hobson, The Theory of Functions of a Real Variable (ed. 2, vol. i., 1921) ; Ch. J. de la Vallee Poussin, Cours d'analyse infinitesimale (1909, 1912) and Integrales de Lebesgue, etc. (1916). See also E. Borel, Lemons sur la theorie des fonctions (ed. 2, 1914), and Lemons sur les fonctions de variables reelles (1905); H. Lebesgue, Lemons sur V integration (1904) and Lemons sur les series trigonometriques (1906) ; C. Caratheodory, Vorlesungen uber reelle Funktionen (1918) ; H. Hahn, Theorie der reellen Funktionen (1921).
4. Integral Equations. Among the remaining developments of modern analysis, perhaps the most remarkable are in the theory ol integral equations. The typical integral equation is
f(x) =
-d)
where f(x) and K(x,t) are given and the unknown function (t\ is to be determined. This equation is called an integral equation ol the first kind ; but it has been found that equations of the form
known as equations of the second kind, are better adapted for the foundation of a general theory. It was shown by I. Fredholm that, if /and K satisfy certain conditions, there is in general one and only one continuous solution (t); the exceptions arise when X is a zero of a certain transcendental function D(X). When X has one of these exceptional values, the equation
has a continuous solution other than the obvious solution 0(0 =o otherwise this is the only solution. The theory has been widely developed by Fredholm, D. Hilbert, V. Volterra and other writers.
See M. Bdcher, An Introduction to the Study of Integral Equations (1909); T. Lalesco, Introduction a la theorie des equations integrate* (1912); H. B. Heywood and M. Frechet, L'fiqtialion de Fredholm et ses applications a la physique mathcmatique (1912) ; V. Volterra, Lemons sur les equations integrates (1913); D. Hilbert, Crundziige einer allgemeinen Theorie der linearen Integralgleichungen (1912); A. Kneser, Die Integralgleichungen und Hire Anwendungen in der Mathematischen Physik (1911); and the third volume of E. Goursat's Cours d 'Analyse (ed. 2, 1915). (G. H. H.)
(5.) GEOMETRY
General remarks will be offered here in regard to two aspects of geometry (see 11.675) which may be held to be of contempo- rary interest, under the headings (a) Foundations of geometry; (b) Theory of classes of surfaces. Under the former heading it is not intended to discuss in detail the so-called Axioms of Geom- etry, for which the reader may be referred to the article with that title (see 11.730), but only to advert in general terms to quest- ions which have indirectly been much in evidence of late in connexion with Einstein's theory of Relativity (see RELATIVITY). Under the second heading is included a quite technical theory, which now has great importance and a developing character.
