Generalized Fatou lemma. Assume that \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{L}^{1}(\mu)\). Prove the following.

(i) If \(u_{n} \geqslant v\) for all \(n \in \mathbb{N}\) and some \(v \in \mathcal{L}^{1}(\mu)\), then

\[\int \liminf _{n ightarrow \infty} u_{n} d \mu \leqslant \liminf _{n ightarrow \infty} \int u_{n} d \mu .\]

(ii) If \(u_{n} \leqslant w\) for all \(n \in \mathbb{N}\) and some \(w \in \mathcal{L}^{1}(\mu)\), then

\[\limsup _{n ightarrow \infty} \int u_{n} d \mu \leqslant \int \limsup _{n ightarrow \infty} u_{n} d \mu\]

(iii) Find examples that show that the upper and lower bounds in (i) and (ii) are necessary. [ mimic and scrutinize the proof of Fatou's lemma, especially when it comes to the application of Beppo Levi's theorem. What goes wrong if we do not have this upper/lower bound? Note that we have an 'invisible' \(v=0\) in Theorem 9.11.]

Data from theorem 9.11

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(Fatou) Let (Un)neNCM(A) be a sequence of positive measur- able functions. Then u:=lim info Un is measurable and lim infundu