Exercise 7.3.6

Question

Let $\left\{r_{1}, r_{2}, r_{3}, \ldotsight\}$ be an enumeration of all the rationals in $[0,1]$, and define
$$
g_{n}(x)= \begin{cases}1 & 	ext { if } x=r_{n} \ 0 & 	ext { otherwise. }\end{cases}
$$
\begin{enumerate}[label=(\alph*)] 
\item Is $G(x)=\sum_{n=1}^{\infty} g_{n}(x)$ integrable on $[0,1]$ ?
\item Is $F(x)=\sum_{n=1}^{\infty} g_{n}(x) / n$ integrable on $[0,1]$ ?
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item $G(x)$ is Dirichlet's function and is not integrable
\item The same approach as that used in Exercise 7.3.2 can be used; in particular the set of points $\{x \in[0,1]: F(x) \geq \epsilon / 2\}$ is finite.
 \end{enumerate}

Exercise 7.3.6

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Exercise 7.3.6

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