Given that $f:[0,1]\to[0,\infty)$ in $L^1$ such that $\int_E f$ $dm\leq\sqrt{m(E)}$ for every $E\subseteq[0,1]$ measurable, prove that $f\in L^p[0,1]$ for all $p\in[1,2)$.
This is a qualifying preparation exam question. None of us have seen anything with $L^p$ spaces before and so have no real idea how to get started. Hints? Suggestions? Proofs? Many thanks.