Let $S$ be a closed subspace of $L^1_m[0, 1]$ such that each $f\in S$ belongs to some $L^p_m[0, 1]$ with $p>1$. Prove that there exists a $p_0 > 1$ such that all $S$ is contained in $L^{p_0}_m[0,1]$.
My attempt: Since for finite measure spaces if $1\leq p\leq q<\infty$, $L_q\subseteq L_p$. So we can take the infimum of all such $p$. But I am not sure how to use the closeness or completeness of $S$ to say that the infimum is strictly greater than $1$.
And what is the counterexample when $S$ is not closed?
