To every subset A \subseteq \mathbb{R} we assign a value m(A) called the Lebesgue outer measure. In the sense that m([0,1]) = 1, it extends the length of an interval and measures the "size" of a general set of real numbers.

However, it is known that there are two disjoint sets A and B such that m(A \cup B) \neq m(A) + m(B). This means that additivity, which is a basic property one would want from any notion of size, can fail, and this is a problem.

The solution is that we give up asking for additivity on all sets. Instead, we take the view that it is enough for additivity to hold on the sets obtained inductively from ordinary sets by ordinary operations. This is the idea of measurable sets. In fact, we prove that additivity does hold on measurable sets.

We also show that the Lebesgue outer measure behaves well with respect to limiting operations. This is one of the main reasons why it becomes so useful later in integration theory.