Section E.1 Lebesgue Integrability Criterion

We record here a useful criterion for Riemann integrability that goes far beyond continuity and monotonicity. A proof can be found in [9].

Definition E.1.1.

A set \(A \subseteq \R\) has measure zero if for every \(\varepsilon \gt 0\) there exists a sequence of open intervals \((I_k)_{k=1}^{\infty}\) such that

\begin{equation*} A \subseteq \bigcup_{k=1}^{\infty} I_k \qquad\text{and}\qquad \sum_{k=1}^{\infty} |I_k| \le \varepsilon, \end{equation*}

where \(|I_k|\) denotes the length of \(I_k\text{.}\) A set of measure zero is also called a null set.

A property is said to hold almost everywhere on an interval if it fails only on a null set.

Example E.1.2.

The set \(\N\) is a null set. Indeed, let \(\varepsilon \gt 0\text{.}\) For each \(k \in \N\text{,}\) define

\begin{equation*} I_k=\left(k-\frac{\varepsilon}{2^{k+1}}, k+\frac{\varepsilon}{2^{k+1}}\right). \end{equation*}

Then \(k \in I_k\) for every \(k\text{,}\) so \(\N \subseteq \bigcup_{k=1}^{\infty} I_k\text{.}\) Also,

\begin{equation*} |I_k|=\frac{\varepsilon}{2^k}, \end{equation*}

and therefore

\begin{equation*} \sum_{k=1}^{\infty} |I_k| = \sum_{k=1}^{\infty} \frac{\varepsilon}{2^k} = \varepsilon. \end{equation*}

Corollary E.1.3.

Every countable subset of \(\R\) is a null set.

Proof.

Let \(A=\{a_1,a_2,\dots\}\) be countably infinite. Given \(\varepsilon \gt 0\text{,}\) put

\begin{equation*} I_k=\left(a_k-\frac{\varepsilon}{2^{k+1}}, a_k+\frac{\varepsilon}{2^{k+1}}\right). \end{equation*}

Then \(A \subseteq \bigcup_{k=1}^{\infty} I_k\) and

\begin{equation*} \sum_{k=1}^{\infty} |I_k| = \sum_{k=1}^{\infty}\frac{\varepsilon}{2^k} = \varepsilon. \end{equation*}

Finite sets are treated in the same way, by allowing all but finitely many intervals to be empty.

Proposition E.1.4.

The union of two null sets is null.

Proof.

Let \(A\) and \(B\) be null sets, and let \(\varepsilon \gt 0\text{.}\) Choose open intervals \((I_k)\) covering \(A\) with

\begin{equation*} \sum_{k=1}^{\infty}|I_k| \le \frac{\varepsilon}{2}, \end{equation*}

and open intervals \((J_k)\) covering \(B\) with

\begin{equation*} \sum_{k=1}^{\infty}|J_k| \le \frac{\varepsilon}{2}. \end{equation*}

Then the countable family consisting of all the intervals \(I_k\) and \(J_k\) covers \(A \cup B\text{,}\) and the total sum of their lengths is at most \(\varepsilon\text{.}\) Hence \(A \cup B\) is null.

Theorem E.1.5. Lebesgue Integrability Criterion.

A bounded function \(f \colon [a,b] \to \R\) is Riemann integrable if and only if it is continuous almost everywhere on \([a,b]\text{;}\) that is, if and only if the set of discontinuities of \(f\) has measure zero.

The Thomae function from Example 7.2.8 illustrates this criterion well: it is continuous at every irrational point and discontinuous at every rational point. Since \(\Q\) is countable, Corollary E.1.3 shows that the set of discontinuities has measure zero, so the theorem predicts that the Thomae function is Riemann integrable.

The criterion also gives a quick explanation for some earlier results. If \(\phi\) is continuous and \(f\) is integrable, then every discontinuity of \(\phi \circ f\) must already be a discontinuity of \(f\text{,}\) so \(\phi \circ f\) is integrable as well. Similarly, if \(f\) and \(g\) are integrable, then every discontinuity of \(fg\) lies in the union of the discontinuity sets of \(f\) and \(g\text{.}\) By Proposition E.1.4, that union has measure zero, so \(fg\) is integrable.

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