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Lebesgue's Dominated Convergence Theorem states:

Assume $g: X \to \overline{\mathbb{R}}$ is a nonnegative, integrable function and that $(f_n)$ is a sequence of measurable functions converging pointwise to f. If $|f_n|≤g$ for all $n$, then $$\lim_{n\to\infty}\int f_n d\mu=\int fd\mu$$

Is the "opposite" true, that is if $\lim_{n\to\infty}\int f_n d\mu\neq\int f d\mu$ and $|f_n|≤g$ is $g$ not integrable?

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    $\begingroup$ The contrapositive has the same truth value as the original statement, so... $\endgroup$
    – Xander Henderson
    Feb 20, 2021 at 14:11

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If $g: X \to \overline{\mathbb{R}}$ is a nonnegative, integrable function and that $(f_n)$ is a sequence of measurable functions converging pointwise to $f$ and if $|f_n|≤g$ for all $n$, then yes since your statement is just the contrapositive, hence equivalent.

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