0.1The Riemann Integral

Updated 2 Sep 2026Chapter (PDF)

This is not a section in Folland and content is drawn from a variety of places, such as Chapters 1-2 of MIRA.

We redefine the Riemann integral and basic results about it.

[0.1.1]Definition(Partition, Refinement of Partition)#

Let $[a, b]$ be a closed interval in $\bR$ . A partition of the interval $[a, b]$ is a list of finitely many numbers $P = x_0, \dots, x_n$ such that

a = x_0 < x_1 < \dots < x_{n - 1} < x_n = b.

We say a partition $P'$ is a refinement of $P$ if $P'$ is a sublist of $P$ . That is, $P'$ contains all of $P$ and more.

We note that if $P = x_0, \dots, x_n$ is a partition of $[a, b]$ , we have that

[a, b] = [x_0, x_1] \cup \cdots \cup [x_{n - 1}, x_n].

A refinement $P' = x_0, x_{0.5}, x_1, \dots, x_{n - 0.5}, x_n$ would give a finer partition of $[a, b]$ as such:

[a, b] = [x_0, x_{0.5}] \cup \cdots \cup [x_{n - 0.5}, x_n].

[0.1.2]Definition(Lower and Upper Riemann Sum)#

Suppose $f: [a, b] \to \bR$ is a bounded function together with a partition $P$ of $[a, b]$ . We define the lower and upper Riemann sums of $f$ on $[a, b]$ as such.

L(f, P, [a, b]) = \sum_{k = 1}^n (x_k - x_{k - 1}) \inf_{[x_{k - 1}, x_k]} f,

U(f, P, [a, b]) = \sum_{k = 1}^n (x_k - x_{k - 1}) \sup_{[x_{k - 1}, x_k]} f.

[0.1.3]Theorem(Lower Riemann Sums Smaller Than Upper Riemann Sums)#

Let $f: [a, b] \to \bR$ be any bounded function together with any two partitions $P, P'$ of $[a, b]$ . Then

L(f, P, [a, b]) \le U(f, P, [a, b]).

[0.1.4]Definition(Lower and Upper Riemann Integral)#

Suppose $f: [a, b] \to \bR$ is a bounded function. The lower Riemann integral and upper Riemann integral of $f$ are defined by

L(f, [a, b]) = \sup_P L(f, P, [a, b]),

U(f, [a, b]) = \inf_P U(f, P, [a, b]).

That is, the lower Riemann integral is the supremum of the lower Riemann sums over all possible partitions of $[a, b]$ . Conversely, the upper Riemann integral is the infimum of the upper Riemann sums over all possible partitions of $[a, b]$ . We take the supremum for the lower Riemann integral since the lower Riemann sums are always an underapproximation of the area under $f$ (and refining makes it more precise), so taking the supremum over all partitions gives us the best approximation for the area under $f$ . Vice-versa is also true.

[0.1.5]Theorem#

The lower Riemann integral is less than or equal to the upper Riemann integral.

[0.1.6]Definition(Riemann Integral)#

A bounded function $f: [a, b] \to \bR$ is said to be Riemann integrable if its lower Riemann integral equals its upper Riemann integral. We define it by

\int_a^b f(x) \, \dd x = L(f, [a, b]) = U(f, [a, b]).

[0.1.7]Theorem(Continuous Function is Riemann Integrable)#

Suppose $f: [a, b] \to \bR$ is continuous. Then it is Riemann integrable.

Proof.

Recall that a continuous map from a compact set $[a, b]$ into $\bR$ is uniformly continuous. Let $\epsilon > 0$ . Then there is some $\delta > 0$ such that

d(x_1, x_2) < \delta \implies d(f(x_1), f(x_2)) < \epsilon.

Let $n \in \bZ_+$ be defined such that $(b - a)/n < \delta$ , and form a partition $P = x_0, \dots, x_n$ of $[a, b]$ where $x_k - x_{k - 1} = (b - a)/n$ . Then note

\begin{align*} U(f, [a, b]) - L(f, [a, b]) &\le U(f, P, [a, b]) - L(f, P, [a, b]) \\ &= \frac{b - a}{n} \sum_{k = 1}^n \left( \sup_{[x_{k - 1}, x_k]} f - \inf_{[x_{k - 1}, x_k]} f \right) \\ &\le (b - a) \epsilon. \end{align*}

Note that $U(f, [a, b]) \le L(f, [a, b]) + (b - a)\epsilon \le L(f, [a, b])$ . Since the lower Riemann integral is always less than or equal than the upper Riemann integral, the two integrals are forced to be equal.

[0.1.8]Theorem(ML Inequality)#

Suppose $f: [a, b] \to \bR$ is Riemann integrable. Then

(b - a) \inf_{[a, b]} f \le \int_a^b f(x) \, \dd x \le (b - a) \sup_{[a, b]} f.

Proof.

Let $P = a, b$ be the trivial partition. Then note

(b - a) \inf_{[a, b]} f = L(f, P, [a, b]) \le L(f, P, [a, b]) = \int_a^b f(x) \, \dd x.

The second inequality follows immediately.