Measure
by Jimmy Tseng
Measure, a mathematical concept, is a function that assigns a number to a set. It is defined in analysis, one of the major branches of modern mathematics. Its use in mathematics and many other fields of study, such as theoretical physics, is common and widespread.
Introduction
Measure informs analysis and is one of the key building blocks of the modern theory of analysis and probability. Measure is the concept on which the standard, modern theory of integration, Lebesgue integration, is defined. Any theory that uses integration will most likely involve measure. Examples of such theories that involve integration and measure are probability, Fourier analysis, differential equations, and others. Any application that requires integration will also most likely involve measure. Examples of such applications that involve integration and measure are the use of the Dirac delta “function,” a mathematical object ubiquitously used in engineering and physics that has some of the behaviors of functions, but does not satisfy the mathematical definition of a function, the need for setting up a way to integrate on exotic spaces, and others. These lists of examples are seemingly inexhaustible.
The Motivation for Measure
One of the major drawbacks of the theory of integration taught to secondary pupils and college first-years in beginning calculus courses is its inability to handle sets that are not the countable unions and countable intersections of intervals. This theory of integration, whose origins lie in the seventeenth century work of Isaac Newton and Gottfried Wilhelm Leibniz, is called Riemann integration. This drawback of Riemann integration encouraged the investigations that lead to measure.
The challenge of handling more general sets requires a theory that can also handle the familiar sets, the countable unions and countable intersections of intervals, in the way already done. The old way of handing intervals is by length. Hence, [1, 4] has length 3; in general, the interval [a, b] has length b – a. There are three properties of length of note:
Length Property 1: The length of [0, 1] is 1.
Length Property 2: For a set that has length, any translation of this set has the same length.
Length Property 3: For a finite or countably infinite sequence of sets that have length and are disjoint from each other, the length of the union of these sets is the sum of the lengths of these sets.
Property 1 is clear since it follows from the very definition of length. The gist of Property 2 can be gleaned from the intervals [0, 1] and [2, 3], which have the same length. [2, 3] is translated by 2 from [0, 1]. On the real number line, this translation is accomplished by adding 2 to every point in [0, 1]. In general, translations on the real number line are accomplished by adding a constant real number to the set to be translated. An example of Property 3 is that the length of [0, 2] is the sum of the lengths of [0, 1] and [1, 2]. To see an example of Property 3 for a countably infinite sequence of sets that have length and are disjoint from each other, consider breaking the interval [0, 1] using the following process. Break [0, 1] in half to make [0, 1/2] and [1/2, 1]. Keep the first piece and break the second piece in half again to make [1/2, 3/4] and [3/4, 1]. Repeat this process to make a sequence of disjoint sets that begin: [0, 1/2], [1/2, 3/4], [3/4, 7/8]…. Note that each set in the sequence is a half of the length of the set before it. From this sequence of sets, a sequence of lengths is inferred: 1/2, 1/4, 1/8…. This sequence of lengths is a geometric sequence whose sum converges to 1. Note that the disjoint sets in the sequence of sets come from breaking up [0, 1]. Hence the union of these disjoint sets is [0, 1], which has length 1. This result agrees with Property 3. To generalize length to arbitrary subsets on the real number line, the mathematician requires that the new theory preserve the three properties of length above.
The Problem of Generalization to Arbitrary Subsets
Unfortunately, a problem arises when length is generalized to arbitrary subsets of the real number line. There are subsets, strange sets to be sure, that cannot be assigned a length. One way of constructing such subsets is to consider the following equivalence relation on the interval [0, 1). Let x and y be two real numbers in [0, 1). Define x to be equivalent to y if and only if x – y is a rational number. To ensure that this relation is a well-defined equivalence relation, one must check that it satisfies the three properties of equivalence relations: x is equivalent to itself; if x is equivalent to y, then y is equivalent to x; and if x is equivalent to y and y is equivalent to z, then x is equivalent to z. Since x – x = 0 is rational, x is equivalent to x. If x – y is rational, then so is y – x. Hence y is equivalent to x if x is equivalent to y. If x – y is rational and y – z is rational, then x - z is also rational. Hence if x is equivalent to y and y is equivalent to z, then x is equivalent to z. The properties of the rational numbers used in the above are that the negation and sums of rational numbers are still rational. Finally there is a theorem that states that an equivalence relation partitions the set, in this case [0, 1), into a disjoint union of equivalence classes, in this case subsets of [0, 1) such that any two elements in the same equivalence class has a rational difference.
Let N be the subset of [0, 1) that contains exactly one element from each equivalence class. Translate N by r, a rational number between 0 and 1, including 0, but excluding 1. In the resultant set, take the subset of the elements that are 1 or larger and further translate the subset of these elements by –1. Call this new set Nr. (Note that Nr is a subset of [0, 1).) As stated above, these translations are just the addition of r and –1 to each element of their respective sets. If x is an element of N, then x + r or x + r – 1 (but not both) is in [0, 1) and in Nr. Any element of an equivalence class is some r or r – 1 away from the element of the same equivalence class in N. Consider the equivalence class with 1/4 and 3/4 in it. If 1/2 is added to 1/4, the result is 3/4. If 1/2 is added to 3/4, the result is 5/4. If 1 is subtracted from 5/4, the result is again contained in [0, 1). Hence, every element of each equivalence class is in Nr for some r. Since the disjoint union of the equivalence classes is [0, 1), [0, 1) lies in the union of Nr. Finally, it is necessary to show that for any two different rational numbers in [0, 1), r and s, Nr and Ns are disjoint. If y is in both, then y – r or y – r + 1 and y – s or y – s + 1 are in N. Since all of these numbers merely differ by a rational number, they must lie in the same equivalence class. Since N is constructed with only one element from each equivalence class, these numbers must be the same number. If y – r = y – s, then r = s. If y – r = y – s + 1, then s = r + 1. This case is impossible since s is less than 1, but r is greater or equal to than 0. Hence s = r + 1 is extraneous and discarded. If y – r + 1 = y – s, then r = s + 1. This case is similarly extraneous and discarded. If y – r + 1 = y – s + 1, then r = s. Hence, r = s. This result implies that if two Nr’s share one element, they share all elements. Therefore, [0, 1) is the disjoint union of the Nr’s for all r, a rational in [0, 1).
What length should be assigned to Nr? By Property 2 of length, the length of Nr for each r must be the same since the Nr’s are just translations of N. By Property 3 of length, the length of [0, 1) must be the sum, over the countably infinite number of rational numbers in [0, 1), of the length of the Nr’s. One can try to assign to each Nr a positive length. Then all the Nr’s have this length. Consequently the sum is infinite. This result implies that the length of [0, 1) is infinite in contradiction to Property 1 of length. (Note that the length of a single point is zero. Hence the length of [0, 1] equals the length of [0, 1).) Thus, the Nr’s cannot have positive length. One can also try to assign to each Nr zero length. Then the sum is zero again in contradiction of Property 1 of length. Given this paradox, the only possible conclusion is that N cannot be assigned any length. Consequently this result shows that some subsets cannot be assigned any length. To keep the three properties of length stated above, the mathematician must restrict the subsets that are assigned a length to certain “nice” ones. One way of constructing nice subsets is to first consider subsets that satisfy Property 3. These are called σ-algebras (σ is the Greek letter that is pronounced like sigma).
σ-algebras
By considering Property 3 in the abstract, the mathematician naturally stumbles on the concept of σ-algebra. As examples of σ-algebras, consider the σ-algebras over the real numbers. One example is the two-element σ-algebra containing the real numbers as an element and the empty set as the other element. Another example is the σ-algebra containing every possible subset of the real numbers. Between these two extremes, there are σ-algebras that the mathematician can generalize length on.
One important σ-algebra on the real numbers is called the Borel σ-algebra on the real numbers. Its elements are the real numbers, the empty set, the closed intervals, countably infinite or finite unions of these closed intervals, and the complements in the real numbers of any of these. (Recall, for example, that the complement in the real numbers of (-∞, 0] is (0, ∞).) Note that length is a function from the Borel σ-algebra to the non-negative real numbers and infinity with the length of the real numbers as infinity, the length of the empty set as zero, and the length of the interval [a, b] as b - a. Using this length function as a model, the mathematician can define a new concept: measure, a function on a σ-algebra to the non-negative real numbers and infinity.
The Definition of Measure
Let X be the set whose subsets will be assigned a non-negative real number or infinity. Let M be a σ-algebra over X. A measure, μ, is function from M to the non-negative real numbers or infinity such that the following two properties are satisfied:
Measure Property 1: For the empty set, denoted by ø, its value under μ is zero.
Measure Property 2: The value of μ under any finite or countably infinite disjoint union of subsets of X that are also elements of M is equal to the finite or countably infinite sum respectively of the value of μ under each of the subsets.
Note that Measure Property 2 is like Length Property 3.
In standard mathematical notation, the definition of measure, μ, is as follows. Given X and M as above, define μ: M → [0, ∞] such that the following two properties hold:
Measure Property 1: μ(ø) = 0.
Measure Property 2: If {En}1∞ is a sequence of disjoint sets in M, then μ(U1∞En) = ∑1∞ μ(En).
The three-tuple (X, M, μ) is called a measure space. An element of M is called a measurable set.
Examples of Measure
Let X be a set. Let the σ-algebra be the set of all subsets of X. On this σ-algebra there are two useful, easily defined measures: the counting measure and the Dirac measure. The counting measure assigns the number of elements of the set as its value. For sets with infinite elements, it assigns infinity as its value.
As for the Dirac measure, its definition depends on a fixed element in X. Hence each element of X has a corresponding Dirac measure at that element. For each subset of X that contains the fixed element, the Dirac measure has value 1. For all other subsets of X, the Dirac measure has value 0.
Another example of measure is the one already mentioned above. The length function is a measure on the real numbers with the Borel σ-algebra. It is an example of a special type of measures called Borel Measures on the real numbers. The proof that the length function is a measure is, however, omitted.
Coda
The definition of measure in the above abstract and general form allows the mathematician to consider a wide array of different sorts of measures, some of which were not recognized to be related to one another and others of which are motivated in part by the definition of measure itself, in a clear, general, unifying abstraction. The study of measure, measure theory, allows the mathematician to consider length as a measure. In identifying length to be a measure, it became possible to generalize and extend length to more sets by utilizing measure theory’s power to avoid the problem mentioned above in extending length. The result is the creation of the Lebesgue measure on the real numbers, the name given to the generalized length function that motivated this discussion of measure. A subset of the real numbers on which one can apply Lebesgue measure is called Lebesgue measurable. The Lebesgue measurable sets form a σ-algebra that is slightly larger and includes the Borel σ-algebra on the real numbers, the domain of the usual length function. Lebesgue measure, then, subsumes length.
Another, even more important, result of measure is its central and crucial role in the refurbishing of the traditional theory of integration, Riemann integration, into the more powerful theory of Lebesgue integration. Measure allows the expansion of the definition of the integral to functions whose domain is any arbitrary set with a corresponding σ-algebra and measure. Just this one generalization is of tremendous import to the theory of integration. These are but a few of the important advances spurred on by the advent of measure.
The study of measures, measure theory, forms a subfield of analysis. Tomes have been written on measures and their applications to many fields of inquiry both inside and outside of mathematics. Current research that involves measures is still in progress. Truly, measure deserves a place as one of the central concepts of modern mathematics.
A Note on the Book Resources
The Principles of Mathematical Analysis is listed to provide the reader with an elementary reference whose content and presentation is slightly beyond a traditional elementary calculus course. Walter Rudin’s book will provide some of the background necessary to read Real Analysis: Modern Techniques and Their Applications where measures are introduced quickly. However, a careful, patient mathematical tyro who closely and carefully reads or those with some practice in reading and understanding mathematics should be able to skip Rudin’s book and go directly to Folland’s book.
Copyright © 2003 by Jimmy Tseng
Book Resources On MeasureThe Principles of Mathematical Analysis by Walter Rudin
Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland
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