Around the turn of the century, mathematicians such as Emile Borel and Henri Lebesgue were looking for a precise definition of the measure of a subset of the real n-dimensional space Rn. Four considerations bounded their search. First, the measure of any 'normal' object such as a line segment in R1, a square in R2, or a cube in R3 should be equal to its length, area, or volume respectively. Second, if A and B are disjoint subsets then the measure of A union B should be equal to the measure of A plus the measure of B. Third, if A and B are congruent sets then their measures should be equal. Fourth, it should be possibly to measure 'most' sets or at least all 'nice' ones. In 1924, Stefan Banach and Alfred Tarski dramatically demonstrated that not all subsets could be measurable if the first three considerations held true. They showed that a ball is equal to the union of six disjoint pieces that can be reassembled into two balls, both of the same volume. Thus the pieces must be non-measurable.
A fifth consideration aided the development of measure theory; if A is a subset of B, then the measure of A should be less than or equal to that of B. Since n-dimensional rectangles have to be measurable by the first consideration, any subset that can be approximated by disjoint unions of rectangles should be measurable too. The measure of an open set (see the article on topology for definitions) is defined as the supremum over all measures of disjoint unions of rectangles contained within the open set. The measure of a compact set is the infimum over all measures of open sets that contain the compact set. The inner measure of any subset is the supremum over all measures of compact sets contained in the subset. The outer measure is the infimum over all measures of open sets that contain the subset. A subset S is measurable if its inner measure equals its outer measure. In this case, the measure of S is to defined to be its inner (or outer) measure.
Lebesgue showed that the set of all measurable subsets is a sigma-algebra. This means that the empty set and the whole space are measurable, complements of measurable sets are measurable, and denumerable unions of measurable sets are measurable. He also proved that the measure of a denumerable disjoint union of measurable sets is equal to the infinite sum of the measures of the sets. Suppose that X and Y are measurable and that Y is contained in X. Then, the measure of X - Y is equal to the measure of X minus the measure of Y. Nowadays, an (abstract) measure on a space is defined to be a sigma algebra of measurable sets and a function from the sigma algebra that satisfies all the above properties. This generality enables the basic ideas of measure theory to be applied to a variety of objects such as groups, manifolds, and function spaces. The principle use of measure theory, however, is Lebesgue integration. Lebesgue integration has the virtue that it can be applied to non-continuous yet 'measurable' functions. A function f is measurable if for every open set V is the range of f, f-1 (V) is measurable.
Here is a non-measurable set. Let E be a subset of the real numbers with the following two properties. First, for any real number x there is a y in E such that x - y is rational. Second, if x and y are both in E and not equal then x - y is irrational. The existence of a set with these properties can be proved with Zorn's lemma. Since any number is arbitrarily close to a rational number, the set E can be chosen so that it is contained in the interval [0,1]. Suppose for a contradiction that E is measurable. For any rational number r, E + r is the set of all numbers of the form x + r where x is an element of E. So, E + r is disjoint from E and congruent to E. The union U, of all sets E + r for all rational r in the interval [-1,1] is contained in [-1,2]. Since U is a denumerable union of measurable sets, it is measurable. U contains [0,1] since any y in [0,1] can be written as e + r for some e in E and r in R. But since e is in [0,1], r must be in [-1,1]. So y is in U. So, the measure of U is in between the measure of [0,1], which is 1, and the measure of [-1,2], which is 3. On the other hand, its measure is equal to the sum of the measures of E + r for r in [-1, 1]. This is because if r and s are distinct rational numbers then E + r is disjoint from E + s. Since there are infinitely many rational numbers in [-1, 1], the measure of U must be infinity times the measure of E. This number is either zero or infinity depending on whether the measure of E is zero or positive. But since U contains [0,1] and is contained in [-1, 2], the measure of U must be between 1 and 3. This contradiction shows that E is not measurable.
Measure theory relies on Cantor's theory of infinite sets. Cantor's theory was controversial. In fact, it was referred to as a "disease of which later generations will cure itself of" by Henri Poincaré. On the other hand, measure theory has been so successful that most mathematicians accepted Cantor's theory in order to validate measure theory. Today, both theories are generally accepted and used without reservation.
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