In this chapter,  we will define and study the Lebesgue integral.  The concepts of an upper function and its Lebesgue integral are introduced first. This leads to a discussion of outer measure, measure, and measurable functions. The definition and derivation of the important properties of the Lebesgue integral on the real line, including the Monotone and Dominated Convergence Theorems, are then carried out. The Lebesgue integral on \(n-\) dimensional Euclidean space is discussed, and versions of the Fubini and Tonelli Theorems on the equality of multiple and iterated integrals are established.