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.