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Let $A$ be a measurable set with finite measure, $f_n$ converges to $f$ in measure on $A$. May I conclude that $f$ is finite almost everywhere on $A$?

Personally, I think the answer is "yes". By contradiction, $f$ is not almost everywhere. It is implied that there exists a set $B$ with positive measure such that $f$ is infinite on $B$. Thus, $\vert f_n - f \vert $ is infinite on $B$, for all $n \in \mathbb{N}$. Therefore, for all $\varepsilon > 0$ $$\{x \in A : \vert f_n - f \vert \ge \varepsilon \} \supset B .$$ So, $\mu(\{ x \in A : \vert f_n - f \vert \ge \varepsilon \}) >0.$ This implies that $\lim\limits_{n \to \infty} \mu(\{ x \in A : \vert f_n - f \vert \ge \varepsilon \}) > 0$. Therefore, $f_n$ doesn't converge to $f$ in measure on $A$ which is clearly absurd.

I'm not sure about my conclusion. Please give me some hints regard this.

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The conclusion is false. First observation, a positive sequence can converge to zero ( e.g. $(1/n)$ ). Apart from this, you have essentially proved that for $(f_n)$ to converge in measure to $f$, the positive measure set where $f$ is infinite must be the same (up to a negligible set) as the one in $(f_n)$ for all $n$ from a certain rank. But there is nothing from this preventing $f$ from having a positive measure set where it is infinite. (A trivial example would be the constant sequence $f_n =f$).

And for practical situations this is not a serious matter, as if our limit function has such a set, then woking on the sets where it is infinite is not very interesting.

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  • $\begingroup$ So, f cannot be finite almost everywhere, right? I'm confused about it. $\endgroup$ – FactorY May 17 at 7:32
  • $\begingroup$ Why would that be ? All I am saying is that we do not need $f$ to be bounded almost everywhere. For example, let be $f:\mathbb{N} \to \mathbb{R} \cup [-\infty,\infty]$ with $f(0) = \infty$ and $f(n) = 0$ otherwise and $\mu$ the coutning measure. Then, for $(f_n)$ to converge in measure to $f$, it is necessary that $f_n(0) = \infty$ for all $n \ge N$ for some $N$. $\endgroup$ – nicomezi May 17 at 9:35
  • $\begingroup$ Oh I understand. Thanks. $\endgroup$ – FactorY May 17 at 16:24
  • $\begingroup$ Sorry to bother you again by citing you but do not hesitate to tell if there is something wrong. @FactorY $\endgroup$ – nicomezi May 21 at 8:29

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