19.6.3.3 Convergence (in measure, almost everywhere)

Problem: Bring an example of a sequence of such functions \( f_{n} \), satisfying the conditions of Fatou's lemma, that \[ \lim _{n \rightarrow \infty} \int_{A} f_{n}(x) \mu(d x): \] a) doesn't exist, b) exists and isn't equal to \( \int_{A} \lim _{n \rightarrow \infty} f_{n}(x) \mu(d x) \).