An introduction to real analysis (1.1) - Riemann Integrability

This post will outline the definition of 'Riemann integrability'

The notion of 'Riemann integrability' is one notion of integrability of a function. There exist several other notions, including Darboux integrability, and Lebesgue integrability.

Lets get started.

Definition 0.9 (Partition) - We call $P$ a 'partition' of $[a,b]$ if $P = \{x_0, x_1, x_2, ... , x_{n-1}, x_n\}$ where $x_i < x_{i+1}$ $\forall i \in \{0,1,2,...,n-1\}$ and $x_0 = a$, $x_n = b$

Visually, if you drew out a horizontal line segment and marked endpoints, and then drew several vertical lines through several different places within your line segment, you would have essentially 'created' a partition of the line segment, where each $x_i$ is a number that represents at what location you drew your several vertical lines through.

In order to work with the Riemann integral, we will also need a way of taking some kind of limit, (interestingly, a limit in the formal sense is not necessary in the definition of the Darboux integral). Therefore we will introduce the notion of a 'refinement' of a partition. We will also again see that an intuitive analogue will be easy to construct once the term has been defined.

Refinement - Suppose $P$ is a partition of $[a,b]$. Let $P'$ be a partition of $[a,b]$. We say $P'$ is a refinement of $P$ if $P \subset P'$ . That is $P'$ contains all the points of $P$ and possibly more.

In order to visualise what this looks like, on your previously drawn partitioned line segment, draw some more vertical lines in several different places (perhaps in a different colour). What you have created, is a refinement of your previously created partition.

We need one more definition before we define the Riemann integral:

Definition 1.0 (Mesh Size) - The 'mesh size' of a partition $P$ of $[a,b]$ is defined to be
 $|| P || = $ max$|x_i - x_{i-1}|$ $\forall i \in \{1,2,3,...,n\}$

It is not hard to see that the 'mesh size' is simply the maximum width of any 'division' within a partition, in other words, it is a measure of how 'fine' or how 'course' any partition of $[a,b]$ is.

We are now ready to define the Riemann integral

Definition 1.1 (Riemann Integral) - A bounded function $f$ is said to be Riemann integrable on $[a,b]$ if  $\lim_{||P|| -> 0} \sum_{i=1}^n f(\xi_i)(x_i -x_{i-1}) $ exists. Where $\xi_i$ is any point in $[x_{i-1},x_i]$ and $P = \{x_0, x_1, ..., x_{n-1}, x_n\}$ is a partition of $[a,b]$

If $f$ is Riemann integrable on $[a,b]$, we say that $\lim_{||P|| -> 0} \sum_{i=1}^n f(\xi_i)(x_i -x_{i-1}) = \int_{a}^b f $

The Riemann integral is sometimes handled slightly differently, and the concept of a 'tagged partition' is introduced. As it has been my experience that introducing this concept does more to confuse the reader than to clarify the concept to the reader, this will be left out. 

This definition is also often supplemented with a picture, and keeping in tradition with this a picture will also be presented below here : (Courtesy : https://mathequality.files.wordpress.com/2012/04/x1-x2-x3-fx1-fx2-fx3.gif)

This is a visual example of one of the sums mentioned in Definition 1.1, where in this case $\xi_i = x_{i-1}$. By definition, if we continue to refine the partition shown in the diagram above, our sum (the sum of the areas of the rectangles) should get 'closer and closer' to a particular number, if this is the case, we call $f$ Riemann integrable over $[a,b]$. It should be noted however, that this must be true not only for $\xi_i = x_{i-1}$ but for any arbitrary $\xi_i \in [x_{i-1},x_i]$. Different choices of $\xi_i$ produce different types of sums, but the limit of these sums as we continue to refine our partition further must all yield the same number.