Let $\{f_n\}$ be a sequence of integrable functions on $X$ that is uniformly integrable. Suppose that $f_n\to f$ pointwise a.e. on $X$ and $f$ is measurable. Assume the measure space has the property that for each $\varepsilon \gt 0$, $X$ is the union of a finite collection of measurable sets, each of measure at most $\varepsilon$.
Can I get some hints to prove that $f$ is integrable over $X$?
Attempt:
since $f_n$ is uniformly integrable, there is a $\delta \gt0$ such that for each $n$ $\int_E |f_n|~d\mu \lt 1,$ where $E$ is a measurable subset of $X$. I can use Fatou's Lemma to show that on $E$ $\int_E |f|~d\mu\lt 1.$
Let $X = \bigcup_{n=1}^k E_n$ where $\mu(E_n) \le \varepsilon$ for each $n$. Then I can say $\mu(X) \le k\varepsilon$. I don't know how this is going help.