The Lebesgue Integral

We have previously defined the Riemann and Riemann Stieltjes integral. The idea for those integrals was roughly as follows:

Subdivide the domain of the function (usually a closed, bounded interval) into finitely many subintervals (the partition)
Construct a simple function that has a constant value on each of the subintervals of the partition (the Riemann sum)
Take the limit of these simple functions as you add more and more points to the partition.
If the limit exists, it is called the Riemann integral, and the function is called Riemann integrable.

Now we will take, in a manner of speaking, the "oposite" approach:

Subdivide the range of the function into finitely many pieces.
Construct a simple function by taking a function whose values are those finitely many numbers. Note that there is no mention on how the domain of this function may look like other than it should be a subset of the domain of the original function.
Take the limit of these simple functions as you add more and more points in the range of the original function.
If the limit exists, it is called the Lebesgue integral, and the function is called Lebesgue integrable.

Now that we know the basic idea, we shall proceed with the usual mathematical rigor to define this new concept. First, we need to clarify what we mean by "simple function".

Definition: Simple Function

A function f defined on a set A which is measurable and takes no more than countably many distinct values c₁, c₂, ... , c_n, ... is called a simple function. A simple function can always be writen as a sum:
- f(x) = c_n _{A_n}(x), where A_n = { x: x A, f(x) = c_n }
Recall that _D(x) is the characteristic function of the set D, i.e.
- _D(x) = 1 if x D and 0 otherwise

A simple function looks like a countable sum, so there is the issue of convergence. However, since the values c_n are all distinct, only one characteristic function will be non-zero for every given x.

Now we can already define the Lebesgue integral for simple functions:

Definition: Lebesgue Integral for Simple Function

If f(x) = c_n _{A_n}(x), is a simple function, then the Lebesgue Integral of f is defined as (A_n)

(bgw)