Let $(\Omega, \mathcal{F}, \mu)$ be a measure space. For a measurable function $f: \Omega \rightarrow \mathbb{R}$, and $p \in[1, \infty)$, let $\|f\|_{p}=\left[\mu\left(|f|^{p}\right)\right]^{1 / p}$. Let $L^{p}$ be the space of all such $f$ with $\|f\|_{p}<\infty$. Explain what is meant by each of the following statements:

(a) A sequence of functions $\left(f_{n}: n \geqslant 1\right)$ is Cauchy in $L^{p}$.

(b) $L^{p}$ is complete.

Show that $L^{p}$ is complete for $p \in[1, \infty)$.

Take $\Omega=(1, \infty), \mathcal{F}$ the Borel $\sigma$-field of $\Omega$, and $\mu$ the Lebesgue measure on $(\Omega, \mathcal{F})$. For $p=1,2$, determine which if any of the following sequences of functions are Cauchy in $L^{p}$

(i) $f_{n}(x)=x^{-1} 1_{(1, n)}(x)$,

(ii) $g_{n}(x)=x^{-2} 1_{(1, n)}(x)$,

where $1_{A}$ denotes the indicator function of the set $A$.