State and prove Fatou's lemma. [You may use the monotone convergence theorem.]

For $(E, \mathcal{E}, \mu)$ a measure space, define $L^{1}:=L^{1}(E, \mathcal{E}, \mu)$ to be the vector space of $\mu$ integrable functions on $E$, where functions equal almost everywhere are identified. Prove that $L^{1}$ is complete for the norm $\|\cdot\|_{1}$,

$$\|f\|_{1}:=\int_{E}|f| d \mu, \quad f \in L^{1} .$$

[You may assume that $\|\cdot\|_{1}$ indeed defines a norm on $L^{1}$.] Give an example of a measure space $(E, \mathcal{E}, \mu)$ and of a sequence $f_{n} \in L^{1}$ that converges to $f$ almost everywhere such that $f \notin L^{1}$.

Now let

$$\mathcal{D}:=\left\{f \in L^{1}: f \geqslant 0 \text { almost everywhere }, \int_{E} f d \mu=1\right\} \text {. }$$

If a sequence $f_{n} \in \mathcal{D}$ converges to $f$ in $L^{1}$, does it follow that $f \in \mathcal{D} ?$ If $f_{n} \in \mathcal{D}$ converges to $f$ almost everywhere, does it follow that $f \in \mathcal{D}$ ? Justify your answers.