In the Monotone Convergence Theorem, the Dominated Convergence Theorem and the Fatou's lemma is having Lebesgue Integrable functions (i.e. functions with finite Lebesgue Integral) a necessary condition, i.e. they do not apply to functions with Lebesgue integral potentially equal to $+\infty$ or $-\infty$?
In positive case, do they imply that also the limiting function $f$ is Lebesgue integrable, i.e. with finite Lebesgue integral?