I have this question because expected value is defined only when "Absolute convergence" is satisfied.

But I see that we freely talk about in micro about infinite expected value though Absolute convergence condition is not satisfied. For example, see

here that though Absolute convergence condition is not satisfied, we talk about infinite expected value.

I think if we strictly follow Absolute convergence condition, we never observe infinite or negative infinite expected value.

What should be done in exam? Is Absolute convergence condition just a theoretical frill and not needed in applications? Please help.