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Introduction.-One of the elementary applications of measure theory is, using two dimensional terminology, the determination of areas in the Euclidean plane. Initially, area is defined only for certain rectangular or elementary figares. It is the object of the theory of measure to extend the definition of area to as large a class of subsets of the plane as possible. The set of figures which can be formed by a finite number of rectangles with sides parallel to fixed directions and with vertices with rational coordinates is dense and denumerable This set plays a r6le in the theory of measure analogous to that of the rational numbers in the construction of the real numbers. It is the object of this note to give an abstract formulation for measure theoretical constructions analogous to the various methods of defining real numbers in terms of rationals.'One starts with an abstract system of point sets which has the pertinent properties of the set of rectangular figures in the plane. This set is extended by methods analogous to those by which the real numbers are obtained. The elements of the extended sets so obtained are abstract, that is, they are not point sets, as were the elements of the original set. The next step is the assignment of point sets to these abstract elements. Certain of these point sets may be ordinate sets. To these, functions can be assigned and, by this means, integration2 is introduced. The integral of a non-negative function is the measure, or area, of its ordinate set. In particular, the ordinate set of a step function is a rectangular figure.A real number may be defined as a sequence of nested intervals of rationals. The analogue of this method yields the theory of content and Riemann integration. The analogue of the definition of real numbers by fundamental, or Cauchy, sequences gives Lebesgue measure and integration. The method of cuts does not lend itself readily to interpretation in terms of measure. The method of bounded monotone sequences, when applied first with ascending and then with descending sequences, is, in essence, the classical method for the definition of Lebesgue measure. A precise formulation of the first two of these methods follows.

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