Let $(\Omega, \mathcal{F}, \mu)$ be a measure space and let $1 \leqslant p \leqslant \infty$.

(a) Define the $L^{p}$-norm $\|f\|_{p}$ of a measurable function $f: \Omega \rightarrow \mathbb{R}$, and define the space $L^{p}(\Omega, \mathcal{F}, \mu) .$

(b) Prove Minkowski's inequality:

$$\|f+g\|_{p} \leqslant\|f\|_{p}+\|g\|_{p} \text { for } f, g \in L^{p}(\Omega, \mathcal{F}, \mu), 1 \leqslant p \leqslant \infty$$

[You may use Hölder's inequality without proof provided it is clearly stated.]

(c) Explain what is meant by saying that $L^{p}(\Omega, \mathcal{F}, \mu)$ is complete. Show that $L^{\infty}(\Omega, \mathcal{F}, \mu)$ is complete.

(d) Suppose that $\left\{f_{n}: n \geqslant 1\right\}$ is a sequence of measurable functions satisfying $\left\|f_{n}\right\|_{p} \rightarrow 0$ as $n \rightarrow \infty$.

(i) Show that if $p=\infty$, then $f_{n} \rightarrow 0$ almost everywhere.

(ii) When $1 \leqslant p<\infty$, give an example of a measure space $(\Omega, \mathcal{F}, \mu)$ and such a sequence $\left\{f_{n}\right\}$ such that, for all $\omega \in \Omega, f_{n}(\omega) \nrightarrow 0$ as $n \rightarrow \infty$.