Lebesgue measure

The outer measure of a subset S of the Real Numbers is the infimum of the set of sums of lengths of open intervals whose (possibly infinite) union covers (contains) S. The notation for the outer measure of S is m*(S).

The outer measure of (0,3) is clearly 3. The outer measure of (0,3) union (4,5) is clearly 4. It is less obvious that the outer measure of the rational numbers is 0.

This measure is always greater than or equal to zero. It is also monotonic. That is, if A is a subset of B then m*(A)<=m*(B).

A set S is called measurable if for any subset of the Real Numbers T the sum of the outer measures of T intersected with S and T intersected with the complement of S is equal to the outer measure of T.

The Lebesque Measure is the outer measure restricted to the domain of measurable sets. Understanding it is absolutely necessary to any student of Real Analysis or Measure Theory. The notation for the Lebesque Measure of a set S is m(S).

While The Zealous Nihilist's definition is correct, it's also a bit incomplete. Here's a (hopefully) more complete definition.

Let us first consider all sets to be contained in a bounded interval X, later we shall remove this restriction.

The outer measure m^*(A) of a set A is defined as:

 *
m (A) = inf m(J)
        A⊂J

Or, in other words, the greatest lower bound of the measures of all the subsets J of the set A. The inner measure m_*(A) is defined

                *
m (A) = m(X) - m (X - A)
 *

or equivalently:

m (A) = sup m(K)
 *      K⊂A

In other words the least upper bound of the measures of all the supersets K of A.

A bounded set E is said to be Lebesgue measurable if m^*(E) = m_*(E), and its Lebesgue measure is the common value of the set's inner and outer measures. One obvious property here is that a set E is measurable if and only if its complement with respect to X is also measurable, and m(E) + m(X-E) = m(X).

The restriction given above on the sets being contained in a bounded interval can be removed. Let A be a set which is not necessarily bounded. Let I^(k) be the bounded interval [-k, k], for k an integer, and let A^(k) = A ∩ I^(k). Then the inner and outer measures of A are defined:

 *           *  (k)
m (A) = sup m (A   )
         k      (k)
m (A) = inf m (A   )
 *       k   *

For a possibly unbounded set E to be considered measurable, E^(k) must be measurable for every k, and then m(E) would be

              (k)
m(E) = lim m(E   )
       k→∞

This definition is the same as the one above given for bounded sets.