# "Opposite" of Lebesgue's Dominated Convergence Theorem

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?

• The contrapositive has the same truth value as the original statement, so... Feb 20, 2021 at 14:11

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.