Conditional convergent improper Riemann integral vs. Lebesgue Integral

Question

For all this, I'm thinking on functions defined on $\mathbb{R}$.

I've already read that if a function $f$ is absolute improper Riemann integrable, then $f$ is Lebesgue integrable and both integral values coincide.

I'm asking if there exists an statement like this: "If a function $f$ is conditional improper Riemann integrable, but not absolute improper Riemann integrable, then $f$ is not Lebesgue integrable."

Or on the other hand, I'm looking for a counterexample.

Answer 1

Assume that $f$ is Riemann integrable on any compact intervals. Then, $f$ is also Lebesgue integrable on any compact intervals. Then, $f$ is Lebesgue integrable on $\mathbb R$ if and only if $$ \lim_{n\to \infty} \int_{I_n} |f| < \infty $$ for any increasing sequence of compact intervals $\bigcup_n I_n = \mathbb R$ by monotone convergence theorem.