We would like to study the integral of a real-valued function f : \mathbb{R} \to \mathbb{R}. However, for several reasons, it is better to begin with nonnegative functions f : \mathbb{R} \to [0, \infty] that are allowed to take the value \infty.

One important idea in the Lebesgue integral is to focus on the values taken by a function. Suppose, for example, that f takes only the values \{0,2,3\}. How should its integral be defined? In that case, the integral should be 2 \cdot m(f^{-1}\{2\}) + 3 \cdot m(f^{-1}\{3\}) that is, the sum over the values in the range, where each value is weighted by the measure of the set of points on which the function takes that value. Such a function is called a simple function, and its integral is very easy to understand. To make sense of this quantity, the sets f^{-1}\{2\} and f^{-1}\{3\} must be measurable, but the important point is that we do not need to know their precise shape. It is enough to know their measures. This is what it means to focus on the range rather than on the domain.

Of course, a general function f need not take only finitely many values. But one can approximate a measurable function by a monotonically increasing sequence of simple functions. For example, one may enumerate the nonnegative rational numbers as q_0, q_1, \dots, and let the nth approximation to f be a simple function whose values are among the rational numbers that appear up to stage n. Since the rational numbers are dense in the real numbers, this gives a sufficiently good approximating sequence.

The Lebesgue integral defined in this way satisfies several important convergence theorems. Here we introduce three of them: the monotone convergence theorem, Fatou's lemma, and the dominated convergence theorem.