Lebesgue Measure

The idea of a "measure", intuitively a rule that determines the size of an object, is very important to mathematics. It is used as a baseline for comparing mathematical "objects", usually sets. However, what consistent way is there to assign sizes to sets that varies over such the wide variety of sets found in mathematics?

One approach is called the Lebesgue measure, named after the mathematician Henri Lebesgue, who introduced the concept around 1902. With this measure, the size of a set A is denoted λ(A). The "objects" or sets that can be measured are considered to reside within the a real Euclidean space Rⁿ for some positive integer n. For each dimension n, Rⁿ is the typical flat (Euclidean) space. For n = 1, it is the line, n = 2, the plane, etc. An intuitively easy way to deal with many sets in real space is simply to measure their volumes. For such sets where this is easily done, the Lebesgue measure coincides with what we traditionally calculate as "volume".

The basis for area in n dimensions is found in one dimension, where "volume" is simply length. To a closed interval [a,b] in R¹ (the set of real numbers from a to b, including a and b, see below), the Lebesgue measure assigns the value b - a.

In more than one dimension, we define the Cartesian product of two closed intervals [a,b] and [c,d] in R¹ as the rectangle in R² bounded by the lines x = a, x = b, y = c, and y = d. Its volume, as per the usual practice, is the product of the lengths of the respective intervals; in this case, it is (b - a)(d - c). In equation form, with the Cartesian product indicated by x, λ([a,b]x[c,d]) = (b - a)(d - c).

This approach is extended to three dimensions by considering the Cartesian product of 3 (non-degenerate) closed intervals, which takes the form of a rectangular prism, the volume of this set being the length times the width times the height, as is normal. In n-dimensions, the Cartesian product of n closed intervals I₁, I₂,..., I_n forms an n-dimension rectangular prism with volume |I₁|·|I₂|·...·|I_n|, where |I_i| is the length of the respective interval.

This rectangular prisms in n-dimensions, or rectangular n-prisms, are the building blocks of volume in Rⁿ. Another basic property of the Lebesgue measure is that, if two sets A and B are disjoint, i.e. do not overlap, or more formally, their intersection is empty, then λ(A∪B) = λ(A) + λ(B). This expression holds as long as the measures are defined. More generally, if n disjoint measurable sets are involved:

This formula simply states that the volume of the union of the sets is the sum of their individual volumes. Remarkably, this even applies to infinite unions of sets, as long as the number of sets is countable, or, equivalently, if the collection of sets can be put into a one-to-one correspondence with the natural numbers (for more, see here).

To define volume for a more general set, one approximates the set as an addition of smaller sets, usually n-prisms. To be more precise, for a set A, if a set of disjoint rectangular n-prisms are contained completely within A, then the sum of their volumes is an lower approximation to the volume of A. If a set of rectangular n-prisms (not necessarily disjoint, i.e. there could be some overlap) completely covers the set A in Rⁿ, then it serves as an upper approximation to the volume of A. This method of finding area is called integration.

An illustration of how rectangular n-prisms can provide upper and lower approximation to the volume of a more general set A.

Another property of the Lebesgue measure is invariance under translation and rotation. In other words, if a set A is rotated or moved around in Rⁿ to another set B, then λ(A) = λ(B). On the other hand, if a set is dilated, or expanded, by a factor d, where d is a positive real number, and the dilated set is denoted dA, then

λ(dA) = dⁿλ(A),

where n is the dimension of the ambient space (Rⁿ).

Now, one can consider various sets in Rⁿ and their Lebesgue measures. First, any point, or more precisely, the set containing a single point P, {P}, has measure 0. This is because it can be viewed as a product of intervals each with length zero. Now, any finite set, that is, including a finite number of points, has Lebesgue measure 0, as such a set can be expressed as the union of many sets, each containing one point, which we known to have measure 0. A set with measure 0 is called negligible.

Similarly, the set of integers, or, for higher dimensions, the set of ordered n-tuplets of the form (a₁,a₂,...,a_n) where the a_i are integers, is negligible, as these sets are countable (this is proven here).

Returning to R¹, the set Q of rational numbers, or those numbers expressible as a quotient of two integers, is countable, and therefore negligible, with λ(Q) = 0. This is the first example of a dense set that is negligible. The higher dimension analog of the rational numbers in R¹ is, for each dimension n, the set of n-tuplets (a₁,a₂,...,a_n) where the a_i are rational numbers. For every n, this set is negligible, with Lebesgue measure 0.

However, the Lebesgue Measure does far more than distinguish between countable and uncountable sets. More properties and measures of sets are presented in the next post.

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Sources: Advanced Calculus of Several Variables by C.H. Edwards Jr., Lebesgue Measure on Wikipedia