The Lebesgue integral is one of the most important and powerful tools in mathematical analysis. The Lebesgue integral can be defined in very general settings and the vector space of real or complex valued Lebesgue integrable functions can be organized into a Banach space. In order to define the Lebesgue integral of a function it is necessary to develop the most basic concepts of measure theory.

Let be a nonempty set. A collection A of subsets of is called a -algebra if it satisfies the following conditions:

• (1) both the empty set and the set are elements of A,
• (2) if A ⊆ belongs to A then its compliment ∖ A also belongs to A,
• (3) if A₁, A₂, ... is a countable collection of sets in A then both

The pair (, A) is called a measurable space. Here is an example of a measurable space. Let ℜ be the...