Let $\left(f_{n}: n \in \mathbb{N}\right)$ be a sequence of non-negative measurable functions defined on a measure space $(E, \mathcal{E}, \mu)$. Show that $\liminf _{n} f_{n}$ is also a non-negative measurable function.

State the Monotone Convergence Theorem.

State and prove Fatou's Lemma.

Let $\left(f_{n}: n \in \mathbb{N}\right)$ be as above. Suppose that $f_{n}(x) \rightarrow f(x)$ as $n \rightarrow \infty$ for all $x \in E$. Show that

$$\mu\left(\min \left\{f_{n}, f\right\}\right) \rightarrow \mu(f) .$$

Deduce that, if $f$ is integrable and $\mu\left(f_{n}\right) \rightarrow \mu(f)$, then $f_{n}$ converges to $f$ in $L^{1}$. [Still assume that $f_{n}$ and $f$ are as above.]