[Math] Counter Examples for Dominated Convergence Theorem and Fatou’s Lemma

Question

Is there an example to see why the dominated convergence theorem fails when there is no integrable function dominates the sequence $f_n(x)$?
Also for Fatou's lemma, is there an example where the strict inequality holds?, i.e:
$$\int_X f(x)\text{d}\mu < \liminf_{n\to\infty} \int_X f_n(x) \text{d}\mu$$

Answer 1

Let $X$ be the line with Lebesgue measure and let $f_n(x) = \chi_{[n,n+1]}(x)$. This is a counterexample for both questions.