Let $f_\alpha$ be a family of continuous positive functions $\mathbb R\to \mathbb R$ 
where the index $\alpha$ runs in a compact metric space 
and the map $\alpha\to f_\alpha$ is continuous 
with respect to compact-open topology on the target. Suppose there is a uniform
upper bound on the integrals of $f_\alpha$'s over $\mathbb R$.

<b>Question.</b> Is $\underset{\alpha}{\sup} f_\alpha$ necessarily an integrable function?

<b>Apology.</b> This sure sounds like a homework level question, but after looking at for a while I am not even sure what the answer is.