This is one of the past qual question. Suppose $\phi$ is a real valued measurable function on $\mathbb{R}$ such that, for any $f$ in $L^{1} (\mathbb{R})$, the product $f\phi$ is also in $L^{1} (\mathbb{R})$. To prove $\phi$ is essentially bounded.
Seriously, I do not know where to start. I kind of thought approaching the problem by contradiction. It seem I am going nowhere from there.