Riemann Integration Theory

Riemann integration is the rigorous foundation behind the definite integrals you first learned in calculus. While early exposure focuses on antiderivatives and areas, the true power—and limitations—of the Riemann integral become clear only when you construct it from the ground up using partitions and sums. This theory is not just historical; it frames the essential questions of "what functions can be integrated?" and "why does the Fundamental Theorem of Calculus work?" that lead directly to more advanced concepts like Lebesgue integration.

Partitions, Darboux Sums, and The Integral

The construction begins with a closed, bounded interval $[a,b]$. A partition $P$ of this interval is a finite set of points $\{x_0,x_1,...,x_n\}$ such that $a=x_0<x_1<...<x_n=b$. This divides $[a,b]$ into $n$ subintervals $[x_{i-1},x_i]$. For a bounded function $f$ defined on $[a,b]$, we examine its behavior on each subinterval.

On the $i$-th subinterval $[x_{i-1},x_i]$, we define $m_i=\inf \{f(x):x\in [x_{i-1},x_i]\}$ and $M_i=\sup \{f(x):x\in [x_{i-1},x_i]\}$. The lower Darboux sum for the partition $P$ is then $L(f,P)={\sum }_{i=1}^n{m}_i\Delta x_i$, where $\Delta x_i=x_i-x_{i-1}$. Geometrically, this is the total area of rectangles inscribed under the graph of $f$. Conversely, the upper Darboux sum is $U(f,P)={\sum }_{i=1}^n{M}_i\Delta x_i$, representing the area of circumscribed rectangles.

A key insight is that for any partition $P$, we always have $L(f,P)\le U(f,P)$. More importantly, refining a partition (adding more points) increases the lower sum and decreases the upper sum. This leads us to consider the "best possible" approximations: the upper integral $U(f)={\inf }_{P}U(f,P)$ (the greatest lower bound of all upper sums) and the lower integral $L(f)={\sup }_{P}L(f,P)$ (the least upper bound of all lower sums). For any bounded function, $L(f)\le U(f)$.

We then say a bounded function $f$ on $[a,b]$ is Riemann integrable if the upper and lower integrals coincide. Their common value is defined as the Riemann integral: $${\int }_a^{b}f(x)dx=L(f)=U(f).$$

Key Classes of Integrable Functions

With the rigorous definition in place, we can prove important theorems about which functions are guaranteed to be integrable. Two fundamental classes are continuous functions and monotone functions.

Continuous Functions: If a function $f$ is continuous on the closed interval $[a,b]$, then it is Riemann integrable on $[a,b]$. The proof leverages the fact that a continuous function on a closed interval is uniformly continuous. This means we can force the oscillation $M_i-m_i$ to be arbitrarily small on all subintervals simultaneously by choosing a sufficiently fine partition, which forces $U(f,P)-L(f,P)$ to be small. This condition, that for any $ϵ>0$ there exists a partition $P$ such that $U(f,P)-L(f,P)<ϵ$, is the standard criterion for establishing integrability.

Monotone Functions: If $f$ is monotone (either non-decreasing or non-increasing) on $[a,b]$, then it is Riemann integrable. The proof is elegant: for a monotone function, the supremum and infimum on any subinterval are simply the values at the endpoints. By choosing a partition with subintervals of sufficiently small equal length, the difference between the upper and lower sums telescopes to a quantity that can be made arbitrarily small.

The Lebesgue Criterion for Riemann Integrability

While continuity and monotonicity are sufficient for integrability, they are not necessary. The complete characterization was provided by Henri Lebesgue. The Lebesgue criterion states: A bounded function $f$ on $[a,b]$ is Riemann integrable if and only if the set of its points of discontinuity has Lebesgue measure zero.

A set has Lebesgue measure zero if it can be covered by a countable collection of intervals whose total length is less than any preassigned $ϵ>0$. Finite sets and countable sets (like the rational numbers $\mathbb{Q}$) have measure zero. This criterion explains many phenomena. For example, the Dirichlet function $f(x)=1$ if $x$ is rational and $0$ if $x$ is irrational is not Riemann integrable on any interval because its discontinuity set (all real numbers) does not have measure zero. Conversely, the Thomae's function (continuous at all irrationals and discontinuous at all rationals) is Riemann integrable because its discontinuity set ($\mathbb{Q}$) is countable and thus has measure zero.

Connection to the Fundamental Theorem of Calculus

The theory culminates in connecting this meticulous "sums of rectangles" definition to the powerful, practical tool of antidifferentiation. The Fundamental Theorem of Calculus (FTC) has two parts that bridge differential and integral calculus.

FTC Part 1: If $f$ is Riemann integrable on $[a,b]$ and we define a new function $F$ by $F(x)={\int }_a^{x}f(t)dt$ for $x$ in $[a,b]$, then $F$ is continuous on $[a,b]$. Moreover, if $f$ is continuous at a point $c$ in $(a,b)$, then $F$ is differentiable at $c$ and $F^{\prime }(c)=f(c)$. This part shows that integration (in the Riemann sense) acts as an "inverse" to differentiation, provided the integrand is continuous.

FTC Part 2: If $F$ is any differentiable function on $[a,b]$ such that its derivative $F^{\prime }$ is Riemann integrable on $[a,b]$, then $${\int }_a^b{F}^{\prime }(x)dx=F(b)-F(a).$$ This is the workhorse theorem used to evaluate definite integrals. The requirement that $F^{\prime }$ be integrable is crucial—the theorem fails for derivatives that are not Riemann integrable, which highlights the limitations of the Riemann integral and motivates the development of more general integrals.

Common Pitfalls

1. Assuming All Bounded Functions Are Integrable: Boundedness is a necessary condition for defining Darboux sums, but it is not sufficient. The classic counterexample is the Dirichlet function, which is bounded but whose upper integral is 1 and lower integral is 0 on any interval, making it non-integrable in the Riemann sense.

1. Misapplying the Fundamental Theorem of Calculus Part 2: A common error is to try to evaluate ${\int }_a^{b}f(x)dx$ as $F(b)-F(a)$ when $F^{\prime }=f$ but $f$ is not Riemann integrable. For example, if $F$ is Volterra's function (a differentiable function with a bounded, non-integrable derivative), the FTC Part 2 does not apply because the integral of $F^{\prime }$ does not exist in the Riemann sense.

1. Confusing the Measure of the Discontinuity Set: When applying the Lebesgue Criterion, remember that the condition is "measure zero," not "countable." While all countable sets have measure zero, there are also uncountable sets (like the Cantor set) that have measure zero. A function discontinuous on an uncountable set of measure zero can still be Riemann integrable.

Summary

• The Riemann integral of a bounded function is rigorously defined via the infimum of upper Darboux sums and the supremum of lower Darboux sums, which must be equal for integrability.
• Two key sufficient conditions for Riemann integrability are that the function is continuous on the closed interval or that it is monotone on the interval.
• The Lebesgue Criterion provides the complete characterization: a bounded function is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero.
• The Fundamental Theorem of Calculus connects differentiation and Riemann integration, but its second part requires the derivative to be Riemann integrable, underscoring the theory's limitations.
• Understanding these foundations clarifies why certain pathological functions fail to be Riemann integrable and points the way toward more general integration theories.