In my measure theory course, I've heard the professor say numerous times that
"there is no conditional convergence in Lebesgue integration, it is always absolute."
As an example, he shared that is it quite easy to prove that Riemann and Lebesgue integrals agree when they both exist, on bounded intervals, yet
$$\int_{0}^{\infty} \frac{\sin{x}}{x} \, dx$$ exists in the sense of Riemann, but does not exist in the sense of Lebesgue as
$$\int_{(0,\infty)} \frac{\sin{x}}{x} \, d\mu. $$
"the first accounts for alternation of sign as in an alternating series, while the second does not, and so $f \in \mathcal{R}$ yet $f \notin \mathscr{L}(\mu)$.
Though, I don't understand what is meant by this. What does he mean when he says Lebesgue integration only happens in an absolute sense? Why does Riemann Integration allow for conditional convergence?