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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
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.
Let $h$ be a step function on $[a,b],$ and let $P=\{{x}_{0}<{x}_{1}<...<{x}_{n}\}$ be a partition of $[a,b]$ such that $h\left(x\right)={a}_{i}$ on the subinterval $({x}_{i-1},{x}_{i})$ determined by $P.$
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