Exercise 7.3.5

Question

Provide an example or give a reason why the request is impossible.
\begin{enumerate}[label=(\alph*)] 
\item A sequence $\left(f_{n}\right) \rightarrow f$ pointwise, where each $f_{n}$ has at most a finite number of discontinuities but $f$ is not integrable.
\item A sequence $\left(g_{n}\right) \rightarrow g$ uniformly where each $g_{n}$ has at most a finite number of discontinuities and $g$ is not integrable.
\item A sequence $\left(h_{n}\right) \rightarrow h$ uniformly where each $h_{n}$ is not integrable but $h$ is integrable.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item Let $(r_n)$ be an enumeration of the rational numbers in $[0, 1]$, let $R_n$ be the set of $r_i$ where $i \leq n$, and let
$$f_n(x) = \begin{cases}
    1 & x \in R_n \
    0 & \text{otherwise}
\end{cases}$$
Then $f_n$ has $n$ discontinuities, and $(f_n)$ approaches Dirichlet's function pointwise.
\item Each $g_n$ must be integrable, so by Exercise 7.2.5 $g$ must also be integrable.
\item Letting $d$ be Dirichlet's function, let $h_n(x) = d(x) / n$, with $h(x) = 0$.
 \end{enumerate}

Exercise 7.3.5

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Exercise 7.3.5

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