Fatou's lemma for measures. Let \((X, \mathscr{A}, \mu)\) be a measure space and let \(\left(A_{n}ight)_{n \in \mathbb{N}}, A_{n} \in \mathscr{A}\), be a sequence of measurable sets. We set  

\[
\liminf _{n ightarrow \infty} A_{n}:=\bigcup_{k \in \mathbb{N}} \bigcap_{n \geqslant k} A_{n} \quad \text { and } \quad \limsup _{n ightarrow \infty} A_{n}:=\bigcap_{k \in \mathbb{N}} \bigcup_{n \geqslant k} A_{n}
\]


[check first that \(\mathbb{1}_{\bigcap_{n \in \mathbb{N}} A_{n}}=\inf _{n \in \mathbb{N}} \mathbb{1}_{A_{n}}\) and \(\mathbb{1}_{\cup_{n \in \mathbb{N}} A_{n}}=\sup _{n \in \mathbb{N}} \mathbb{1}_{A_{n}}\).]
(ii) Prove that \(\mu\left(\liminf _{n ightarrow \infty} A_{n}ight) \leqslant \liminf _{n ightarrow \infty} \mu\left(A_{n}ight)\).

(iii) Prove that \(\limsup _{n ightarrow \infty} \mu\left(A_{n}ight) \leqslant \mu\left(\limsup _{n ightarrow \infty} A_{n}ight)\) if \(\mu\) is a finite measure.
(iv) Provide an example showing that (iii) fails if \(\mu\) is not finite.