Let $(E, \mathcal{E}, \mu)$ be a finite measure space, i.e. $\mu(E)<\infty$, and let $1 \leqslant p \leqslant \infty$.

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

In the following you may use inequalities from the lectures without proof, provided they are clearly stated.

(b) Let $f, f_{1}, f_{2}, \ldots \in L^{p}(E, \mathcal{E}, \mu)$. Show that $f_{n} \rightarrow f$ in $L^{p}$ implies $\left\|f_{n}\right\|_{p} \rightarrow\|f\|_{p}$.

(c) Let $f: E \rightarrow \mathbb{R}$ be a bounded measurable function with $\|f\|_{\infty}>0$. Let

$$M_{n}=\int_{E}|f|^{n} d \mu$$

Show that $M_{n} \in(0, \infty)$ and $M_{n+1} M_{n-1} \geqslant M_{n}^{2}$.

By using Jensen's inequality, or otherwise, show that

$$\mu(E)^{-1 / n}\|f\|_{n} \leqslant M_{n+1} / M_{n} \leqslant\|f\|_{\infty}$$

Prove that $\lim _{n \rightarrow \infty} M_{n+1} / M_{n}=\|f\|_{\infty} .$

$\left[\right.$ Observe that $\left.|f| \geqslant 1_{\left\{|f|>\|f\|_{\infty}-\epsilon\right\}}\left(\|f\|_{\infty}-\epsilon\right) .\right]$