Definitions and Examples

Subsection 7.1.1 Partitions and Darboux Sums

A partition of a closed bounded interval \([a,b]\) is a finite subset of \([a,b]\) containing both endpoints. We list a partition in increasing order:

\begin{equation*} P=\{x_0,x_1,\dots,x_n\}, \qquad a=x_0 \lt x_1 \lt \cdots \lt x_n=b. \end{equation*}

For such a partition, we write

\begin{equation*} I_i=[x_{i-1},x_i], \qquad \Delta x_i=x_i-x_{i-1} \qquad (1 \le i \le n). \end{equation*}

For now, we write \(\int_a^b f\) rather than \(\int_a^b f(x)\,dx\text{.}\) At this stage there is no risk of confusion, because the interval and the function being integrated are both fixed. Only later, when we discuss change of variables, will it become useful to display the variable of integration explicitly.

Now let \(f \colon [a,b] \to \R\) be bounded, and let \(P=\{x_0,\dots,x_n\}\) be a partition of \([a,b]\text{.}\) For each subinterval \(I_i=[x_{i-1},x_i]\text{,}\) define

\begin{equation*} m_i=\inf\{f(x):x \in I_i\}, \qquad M_i=\sup\{f(x):x \in I_i\}. \end{equation*}

The numbers

\begin{equation*} L(P,f)=\sum_{i=1}^n m_i \Delta x_i \qquad\text{and}\qquad U(P,f)=\sum_{i=1}^n M_i \Delta x_i \end{equation*}

are called the lower Darboux sum and upper Darboux sum of \(f\) with respect to \(P\text{.}\)

Example 7.1.1.

Let \(f(x)=x^2\) on \([0,1]\text{,}\) and let \(P=\{0,\frac13,\frac23,1\}\text{.}\) Since \(f\) is increasing on \([0,1]\text{,}\) the infimum on each subinterval occurs at the left endpoint and the supremum occurs at the right endpoint. Therefore

\begin{equation*} L(P,f) = \left(0+\frac19+\frac49\right)\frac13 = \frac{5}{27}, \end{equation*}

while

\begin{equation*} U(P,f) = \left(\frac19+\frac49+1\right)\frac13 = \frac{14}{27}. \end{equation*}

A parabola on [0,1] with lower and upper Darboux rectangles for the partition at one third and two thirds. — Figure 7.1.2. The lower Darboux sum, the upper Darboux sum, and the curve \(y=x^2\) for the partition \(\{0,\frac13,\frac23,1\}\text{.}\)

Example 7.1.3.

If \(f(x)=c\) is constant on \([a,b]\text{,}\) then for every partition \(P\) we have \(m_i=M_i=c\) for every \(i\text{.}\) Hence

\begin{equation*} L(P,f)=c(b-a)=U(P,f). \end{equation*}

So lower and upper sums already agree for every partition.

Subsection 7.1.2 Upper and Lower Integrals

Since \(f\) is bounded, there exist real numbers \(m \le M\) such that

\begin{equation*} m \le f(x) \le M \qquad (x \in [a,b]). \end{equation*}

Therefore, for every partition \(P\text{,}\)

\begin{equation*} m(b-a) \le L(P,f) \le U(P,f) \le M(b-a). \end{equation*}

Thus the set of all lower sums is bounded above, and the set of all upper sums is bounded below.

By the The Completeness Axiom of \(\R\), the least upper bound of the set of lower sums exists, and likewise the greatest lower bound of the set of upper sums exists. We define the lower integral and upper integral of \(f\) by

\begin{equation*} \underline{\int_a^b} f := \sup\{L(P,f): P \text{ a partition of } [a,b]\}, \end{equation*}

and

\begin{equation*} \overline{\int_a^b} f := \inf\{U(P,f): P \text{ a partition of } [a,b]\}. \end{equation*}

If \(P\) and \(P'\) are partitions of \([a,b]\text{,}\) we say that \(P'\) is a refinement of \(P\) if \(P \subseteq P'\text{.}\)

Corollary 7.1.5.

For every partition \(P\) of \([a,b]\text{,}\)

\begin{equation*} L(P,f) \le \underline{\int_a^b} f \le \overline{\int_a^b} f \le U(P,f). \end{equation*}

Proof.

Let \(P\) and \(P'\) be any two partitions. Then \(P \cup P'\) is a common refinement of both. By Proposition 7.1.4,

\begin{equation*} L(P,f) \le L(P \cup P',f) \le U(P \cup P',f) \le U(P',f). \end{equation*}

Since \(P'\) is arbitrary, \(L(P,f)\) is a lower bound for the set of upper sums. Hence \(L(P,f) \le \overline{\int_a^b} f\text{.}\) Therefore every upper integral is an upper bound for the set of lower sums, so \(\underline{\int_a^b} f \le \overline{\int_a^b} f\text{.}\) The remaining inequalities follow directly from the definitions of supremum and infimum.

Example 7.1.6.

Define the Dirichlet function \(1_{\Q} \colon [0,1] \to \R\) by

\begin{equation*} 1_{\Q}(x)= \begin{cases} 1 & \text{if } x \in \Q,\\ 0 & \text{if } x \notin \Q. \end{cases} \end{equation*}

Then

\begin{equation*} \underline{\int_0^1} 1_{\Q} = 0 \qquad\text{and}\qquad \overline{\int_0^1} 1_{\Q} = 1. \end{equation*}

Indeed, by Proposition 1.2.7, every nondegenerate subinterval of \([0,1]\) contains a rational number. If \([u,v]\) is such a subinterval, then \([u-\sqrt{2},v-\sqrt{2}]\) is also nondegenerate, so again by Proposition 1.2.7 there is some \(q \in \Q\) with \(u-\sqrt{2} \lt q \lt v-\sqrt{2}\text{.}\) Then \(q+\sqrt{2}\) is irrational and lies in \([u,v]\text{.}\) Thus on each subinterval of any partition the infimum is \(0\) and the supremum is \(1\text{.}\) Hence every lower sum is \(0\) and every upper sum is \(1\text{.}\)

Subsection 7.1.3 Riemann Integrability

A bounded function \(f \colon [a,b] \to \R\) is Riemann integrable if its lower and upper integrals are equal:

\begin{equation*} \underline{\int_a^b} f = \overline{\int_a^b} f. \end{equation*}

The common value is called the Riemann integral of \(f\) on \([a,b]\text{,}\) and is denoted by

\begin{equation*} \int_a^b f. \end{equation*}

We write \(R[a,b]\) for the set of all Riemann integrable functions on \([a,b]\text{.}\)

Proposition 7.1.7. Integrability Criterion.

A bounded function \(f \colon [a,b] \to \R\) is Riemann integrable if and only if for every \(\varepsilon \gt 0\) there exists a partition \(P\) of \([a,b]\) such that

\begin{equation*} U(P,f)-L(P,f) \lt \varepsilon. \end{equation*}

Proof.

First assume that for every \(\varepsilon \gt 0\) there exists a partition \(P\) with \(U(P,f)-L(P,f) \lt \varepsilon\text{.}\) By Corollary 7.1.5,

\begin{equation*} 0 \le \overline{\int_a^b} f - \underline{\int_a^b} f \le U(P,f)-L(P,f). \end{equation*}

Since the right-hand side can be made arbitrarily small, the difference of the upper and lower integrals must be \(0\text{.}\) Hence \(f\) is integrable.

Conversely, suppose \(f\) is integrable, and let \(I=\int_a^b f\text{.}\) Given \(\varepsilon \gt 0\text{,}\) choose partitions \(P'\) and \(P''\) such that

\begin{equation*} U(P',f) \lt I+\frac{\varepsilon}{2} \qquad\text{and}\qquad I-\frac{\varepsilon}{2} \lt L(P'',f). \end{equation*}

Let \(P=P' \cup P''\text{.}\) Then \(P\) refines both \(P'\) and \(P''\text{,}\) so

\begin{equation*} I-\frac{\varepsilon}{2} \lt L(P'',f) \le L(P,f) \le U(P,f) \le U(P',f) \lt I+\frac{\varepsilon}{2}. \end{equation*}

Therefore \(U(P,f)-L(P,f) \lt \varepsilon\text{.}\)

Example 7.1.8.

The function \(f(x)=x\) is Riemann integrable on \([0,1]\text{.}\) For each \(n \in \N\text{,}\) let

\begin{equation*} P_n=\left\{0,\frac{1}{n},\frac{2}{n},\dots,\frac{n-1}{n},1\right\}. \end{equation*}

Since \(f\) is increasing,

\begin{equation*} L(P_n,f) = \frac{1}{n^2}\sum_{k=0}^{n-1} k = \frac{n-1}{2n} \end{equation*}

and

\begin{equation*} U(P_n,f) = \frac{1}{n^2}\sum_{k=1}^{n} k = \frac{n+1}{2n}. \end{equation*}

Hence

\begin{equation*} U(P_n,f)-L(P_n,f)=\frac{1}{n} \to 0. \end{equation*}

By the Integrability Criterion, the function \(x\) is integrable on \([0,1]\text{.}\) Also,

\begin{equation*} \frac{n-1}{2n} \le \int_0^1 x \le \frac{n+1}{2n} \end{equation*}

for every \(n\text{,}\) so letting \(n \to \infty\) gives

\begin{equation*} \int_0^1 x = \frac12. \end{equation*}

Finally, if \(f\) is integrable on \([a,b]\text{,}\) we extend the notation by setting

\begin{equation*} \int_a^a f = 0, \qquad \int_b^a f = -\int_a^b f. \end{equation*}

Section 7.1 Definitions and Examples

Subsection 7.1.1 Partitions and Darboux Sums

Example 7.1.1.

Example 7.1.3.

Subsection 7.1.2 Upper and Lower Integrals

Proposition 7.1.4.

Proof.

Corollary 7.1.5.

Proof.

Example 7.1.6.

Subsection 7.1.3 Riemann Integrability

Proposition 7.1.7. Integrability Criterion.

Proof.

Example 7.1.8.