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)
Or, in other words, the greatest lower bound
of the measure
s of all the subsets J of the set A. The inner measure m*
(A) is defined
m (A) = m(X) - m (X - A)
m (A) = sup m(K)
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 )
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
m(E) = lim m(E )
This definition is the same as the one above given for bounded sets.