Problem 1
Prove that the Dirichlet function $$ f(x)= \begin{cases}1 & \text { if } x \text { is rational } \\ 0 & \text { if } x \text { is irrational }\end{cases} $$ fails to have a Riemann integral over any interval \([a, b]\). Prove that the Lebesgue integral off over any measurable set A exists and equals zero.
Problem 2
Find the Lebesgue integral of the function $$ f(x)= \begin{cases}\frac{1}{q} & \text { if } x=\frac{P}{q} \text { is rational } \\ 1 & \text { if } x \text { is irrational }\end{cases} $$ over the interval \([\mathrm{a}, b]\).
Problem 3
Prove that a) Iff is integrable on a set \(Z\) of measure zero, then $$ \int_{Z} f(x) d \mu=0 $$ b) Iff is integrable on A, then $$ \int_{A^{\prime}} \mathrm{f}(x) d \mu=\int_{A} f(x) d \mu $$ for every subset \(\mathrm{A}^{\prime} \subset A\) such that \(\mu\left(A-\mathrm{A}^{\prime}\right)=0\). Comment. We can regard a) as a limiting case of Theorem \(6 .\)
Problem 6
Let $$ A=\bigcup_{n} A_{n} $$ be a finite or countable union of pairwise disjoint sets \(A\), and suppose \(f\) is integrable on each \(A\), and satisfies the condition $$ \sum_{\hbar} \int_{A_{n}}|f(x)| d \mu<\infty $$ Prove that \(\mathrm{f}\) is integrable on \(A\). Hint. Iff is simple, with values \(y, y_{2}, \ldots\), let the sets \(B_{h}\) and \(B_{n k}\) be the same as in the proof of Theorem 4 . Then $$ \int_{A_{4}}|f(x)| d \mu=\int_{h}\left|y_{k}\right| \mu\left(B_{n k}\right) $$ The absolute convergence of (19) implies the convergence of $$ \sum_{n} \sum_{k}\left|y_{k}\right| \mu\left(B_{n k}\right)=\sum_{k}\left|y_{k}\right| \sum_{n} \mu\left(B_{n k}\right)=\sum_{k}\left|y_{k}\right| \mu\left(B_{k}\right) $$ and hence the integrability off on \(A\). In the general case, let \(g\) be a simple function approximating \(f\), and show that (19) implies the convergence $$ \sum_{\pi} \int_{A_{n}}|g(x)| d \mu $$ so that \(g\), and hence \(f\), is integrable on \(A\). Comment. This is essentially the converse of Theorem \(4 .\)
