Suppose $\mathscr{C}$ is a family of random variables on $(\Omega,\mathscr{F},\mu)$ be $\mathcal{L}^p$-bounded. i.e. \begin{equation*} \sup_{X\in\mathscr{C}} \left\Vert X\right\Vert_p <\infty \end{equation*}
Then is it necessarily true that \begin{equation*} \left\Vert \sup_{X\in\mathscr{C}}\left\vert X\right\vert\right\Vert_p<\infty \end{equation*} If so - could someone please show me a proof? And if not, what would be the weakest sufficient conditions which would make this statement true (and how would you prove it).
Thanks
You say the words "random variables" so I am assuming that $\mu$ is a probability measure, i.e., $\mu(\Omega)=1$.
In this case, your claim is false unless $p=\infty$. When $p=\infty$ it is easily verified to be true.
Let $p<\infty$. We let $\Omega=[0,1]$ with Lebesgue measure.
For $n \in \Bbb N$ and $0 \leq k \leq n-1$ we define $X_{n,k}$ to be $n^{1/p}$ times the indicator function of the interval $[k/n, (k+1)/n]$. Then clearly $\|X_{n,k}\|_p=1$ for all $k$ and $n$. However $\sup_{n,k}|X_{n,k}| = +\infty$ since $n^{1/p} \to \infty$ as $n \to \infty$. Hence $$\big\| \sup_{n,k} |X_{n,k}| \big\|_q = +\infty,\;\;\;\;\;\; \forall q \in [1,\infty]$$
The weakest conditions (of which I know) that would make your claim true is that $p>1$, $\mathscr C = [0,\infty)$ or $\mathscr C = \Bbb N$, and $(X_t)_{t \geq 0}$ is a submartingale. In this case, the result follows from Doob's $L^p$ inequality: $$\big\|\sup_{t \geq 0}|X_t| \big\|_p \leq \frac{p}{p-1} \sup_{t \geq 0} \|X_t\|_p$$