5.1 Integrals of step functions

We begin by defining the integral of certain (but not all) bounded, real-valued functions whose domains are closed bounded intervals.Later, we will extend the definition of integral to certain kinds of unbounded complex-valued functionswhose domains are still intervals, but which need not be either closed or bounded.First, we recall from [link] the following definitions.

Let $[a,b]$ be a closed bounded interval of real numbers. By a partition of $[a,b]$ we mean a finite set $P=\{x_0<x_1<...<x_n\}$ of $n+1$ points, where $x_0=a$ and $x_n=b.$

The $n$ intervals $\left\{[x_{i-1},x_i]\right\}$ are called the closed subintervals of the partition $P,$ and the $n$ intervals $\left\{(x_{i-1},x_i)\right\}$ are called the open subintervals or elements of $P.$

We write $\parallel P\parallel$ for the maximum of the numbers (lengths of the subintervals) $\{x_i-x_{i-1}\},$ and call $\parallel P\parallel$ the mesh size of the partition $P.$

If a partition $P=\left\{x_i\right\}$ is contained in another partition $Q=\left\{y_j\right\},$ i.e., each $x_i$ equals some $y_j,$ then we say that $Q$ is finer than $P.$

Let $f$ be a function on an interval $[a,b],$ and let $P=\{x_0<...<x_n\}$ be a partition of $[a,b].$ Physicists often consider sums of the form

$$S_{P,\left\{y_i\right\}}=\sum _{i=1}^{n}f\left(y_i\right)(x_i-x_{i-1}),$$

where $y_i$ is a point in the subinterval $(x_{i-1},x_i).$ These sums (called Riemann sums) are approximations of physical quantities, and the limit of these sums, as the mesh of the partition becomes smaller and smaller,should represent a precise value of the physical quantity. What precisely is meant by the “ limit” of such sums is already a subtle question,but even having decided on what that definition should be, it is as important and difficult to determine whether or not such a limit exists for many (or even any) functions $f.$ We approach this question from a slightly different point of view, but we will revisit Riemann sums in the end.

Again we recall from [link] the following.

Let $[a,b]$ be a closed bounded interval in $R.$ A real-valued function $h:[a,b]\to R$ is called a step function if there exists a partition $P=\{x_0<x_1<...<x_n\}$ of $[a,b]$ such that for each $1\le i\le n$ there exists a number $a_i$ such that $h\left(x\right)=a_i$ for all $x\in (x_{i-1},x_i).$

REMARK A step function $h$ is constant on the open subintervals (or elements) of a certain partition. Of course, the partition is not unique.Indeed, if $P$ is such a partition, we may add more points to it, making a larger partition having moresubintervals, and the function $h$ will still be constant on these new open subintervals. That is, a given step function can be described using various distinct partitions.

Also, the values of a step function at the partition points themselves is irrelevant. We only require that it be constant on the open subintervals.