the Lebesgue integral is formed by grouping together points of the x-axis at which the function f takes neiboring values. The key notion of Lebesgue integration is to partition the range of a function rather than its domain.--Kolmogorov
There are several equivalent ways' to define the Lebesgue integral and develop it main properties. The approach we have chosen is based on the notation that the integral of a non-negative f should represent the volume of the region under the graph of f.--Wheeden
We shall define on a measurable space X, in which R is an o-ring of measureable sets, and u is the measure. The reader who wishes to visualize a more concrete situation may think of X as the real line or an interval, and of u as the Lebesgue measure m--Rudin 1
Defined it as the "undergraph"--Chapman
In what follows, (X,F,m) is a space with a -field of sets, and m a measure on F. The purpose of this chapter is to develop the theory of the Lebesgue integral for functions defined on X. The theory starts with simple functions, that is functions which take on only finitely many non-zero values, say {a1, . . . , an} and where
In other words, we start with functions of the form
Then, for any E 2 F we would like to define the integral of a simple function
over E as
and extend this definition by some sort of limiting process to a broader class of functions. I haven’t yet specified what the range of the functions should be. Certainly, even to get started, we have to allow our functions to take values in a vector space over R, in order that the expression on the right of (5.2) make sense. In fact, I will eventually allow f to take values in a Banach space. However the theory is a bit simpler for real valued functions, where the linear order of the reals makes some arguments easier. Of course it would then be no problem to pass to any finite dimensional space over the reals. But we will on occasion need integrals in infinite dimensional Banach spaces, and that will require a little reworking of the theory. 133--Sternberg
A useful representation of simple functions is obtained as follows, suppose the function F(A)to R assume the distinct values a1 a2 a3 the there is a partition of A into pairwise disjoint measurable sets Asub p such that the function equals a sub p for x in the element of A sub p--Sprecher