"Opposite" of Lebesgue's Dominated Convergence Theorem

Question

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?

Answer 1

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.