In this post I’ll outline the basic theory of Riemann integration. Let be a bounded function, i.e. there exists such that for all . A **partition** of is a finite subset such that . The **lower sum** of a partition is

The **upper sum** of a partition is

Observe that for any partition , , since for each . Given a partition , we say that is a **partition** of if . Time for a theorem:

Theorem 1If is a refinement of , then and

*Proof:* Suppose and where for some . Then

Since and

it follows that , so that . The proof that is entirely analogous.

We’re now ready to define the integral. If for each there exists a partition such that , then is said to be **Riemann integrable**, and we write

For a basic example, constant functions are integrable. For if for all , then for any partition , .

An equivalent, and easier to use, definition of Riemann integrability is this: is integrable iff there exists a sequence of partitions with and

In this case,

Another example: let , be a sequence of functions on defined by and , where

Then

Clearly is bounded, as for all . Define a sequence of partitions by and for . Then for ,

as , and for ,

Evaluating this sum, we have

It follows that

Since

for all . Hence is integrable, and

Now let’s compute the Lebesgue integral of this function. Since and , by monotone convergence we have