Measure theory: The heart of the matter

Abstract

The art and science of measuremen t -which is the use of a number to describe the amount of some characteristic or quality possessed by an object--constitutes a cornerstone of modern science, and perhaps of all civilized society. Among the earliest qualities which were subjected to systemat ic measu remen t were length, area, and volume. Knowledge of the area of a plot of ground can be used to predict its crop yield or to help estimate the size of an enemy force encamped upon it. Most ancient civilizations developed the basic idea of units of measurement. The Egyptian unit of length, for example, was called the cubit. The Egyptians lived on relatively flat expanses of ground, and it seems that they mostly subdivided their territory into rectangular plots whose areas could be expressed in whole numbers of square cubits. The ancient Greeks, on the other hand, inhabited hilly terrain with limited amounts of flat land available for agriculture. They needed to ascertain, and so to compare, the amounts of area in pre-existing, irregularly shaped plots of ground. This accident of geography, coupled with the ever-present military threat posed by the Persian hordes to the east, is held to be largely responsible for the vastly greater sophistication of Greek mathematics over that of the Egyptians. And measurement is a key feature of their mathematics, not in the modern sense of assigning a number to an object, but in the sense of establishing exact relationships between various quantities (mainly lengths and areas). Perhaps the most famous example is the "a 2 + b 2 = c 2'' relationship of the Pythagorean theorem. The Greeks looked upon this as the equality of two areas. Mathematical measure theory is a branch of modern mathematics which deals with systematic techniques for measuring complicated or irregular objects when the measurements of simple objects are known in advance. Its central idea has a long lineage dating back to a technique invented by the Greeks.