Example of closed strict subspace of $L^p$ that is dense in $L^r$ for $r<p$

Question

This is a Stanford Analysis qual question. Any hints\ solutions\ references are highly appreciated.

Let $\mu$ be the Lebesgue measure on $[0,1]$. For $1<p<\infty$ construct

1. a subspace of $L^p([0,1],\mu)$, which is not dense in $L^p$ but is dense in $L^r$ for all $r<p$.

2. a subspace of $L^\infty([0,1],\mu)$ which is dense in $L^p$ but not in $L^s$ for any $s>p$

I find is interesting that such a thing would hold because:

For 1. by Holder's inequality, for $1\leq q<p\leq \infty$, we have $L^p \subset L^q$.

But of course, the norms are different so I guess what we need is a subspace $M$ closed in $L^p$ but dense in $L^r$ for $r<p$.

Any help will be appreciated. Thanks in advance