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 $\mathbb R$.

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

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