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This post is to clarify if the following example works. Let ${q_n}$ be an enumeration of $\mathbb{Q}$.

For each n define $f_n(x)=1$ if $x\in\{q_1,...,q_n\}$ and $f_n(x)=0$ otherwise. Then for all n $\int f_n dx=0$ while $\int f dx$ does not exist where $lim_{n}f_n=f$ and $f$ is the Dirichlet function.

Is this valid?

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    $\begingroup$ yes it is correct. Notice that a countable enumeration of $\mathbb Q$ is a tautology. $\endgroup$
    – Surb
    Aug 30, 2020 at 9:07
  • $\begingroup$ I always thought enumeration just meant "indexed over some set" but thanks. $\endgroup$
    – Jhon Doe
    Aug 30, 2020 at 9:10
  • $\begingroup$ @JhonDoe Yes, but if "enumeration" means that each elements appears only once, then any enumeration of $\mathbb{Q}$ is countable. $\endgroup$ Aug 30, 2020 at 9:17

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