Lebesgue Measure

How do we measure the ”size” of a set in IR? Let’s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its length, which is used frequently in differentiation and integration. For any bounded interval I (open, closed, half-open) with endpoints a and b (a ≤ b), the length of I is defined by `(I) = b − a. Of course, the length of any unbounded interval is defined to be infinite, that is, `(I) =∞ if I is of the form (a,∞), (−∞, b), or (−∞,∞). How do we measure the size of sets other than intervals? An extension to unions of intervals is obvious, but is much less obvious for arbitrary sets. For instance, what is the size of the set of irrational numbers in [0, 2]? Is it possible to extend this concept of length (or size) of an interval to arbitrary sets? Lebesgue measure is one of several approaches to solving this problem. Here, we want to define and study the basic properties of Lebesgue measure. Given a set E of real numbers, μ(E) will represent its Lebesgue measure. Before defining this concept, let’s consider the properties that it should have.