(a) Let $(X, \mathcal{A}, \mu)$ be a measure space. Define the spaces $L^{p}(X)$ for $p \in[1, \infty]$. Prove that if $\mu(X)<\infty$ then $L^{q}(X) \subset L^{p}(X)$ for all $1 \leqslant p<q \leqslant \infty$.

(b) Now let $X=\mathbb{R}^{n}$ endowed with Borel sets and Lebesgue measure. Describe the dual spaces of $L^{p}(X)$ for $p \in[1, \infty)$. Define reflexivity and say which $L^{p}(X)$ are reflexive. Prove that $L^{1}(X)$ is not the dual space of $L^{\infty}(X)$

(c) Now let $X \subset \mathbb{R}^{n}$ be a Borel subset and consider the measure space $(X, \mathcal{A}, \mu)$ induced from Borel sets and Lebesgue measure on $\mathbb{R}^{n}$.

(i) Given any $p \in[1, \infty]$, prove that any sequence $\left(f_{n}\right)$ in $L^{p}(X)$ converging in $L^{p}(X)$ to some $f \in L^{p}(X)$ admits a subsequence converging almost everywhere to $f$.

(ii) Prove that if $L^{q}(X) \subset L^{p}(X)$ for $1 \leqslant p<q \leqslant \infty$ then $\mu(X)<\infty$. [Hint: You might want to prove first that the inclusion is continuous with the help of one of the corollaries of Baire's category theorem.]