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).