I have two functions $f, g:\mathbb{R}\rightarrow \mathbb{R}_{\ge0}$, that are continuous. I know that $\int\limits_{-\infty}^\infty f(s) \, ds=C_1<\infty$, and $g(s)\le C_2$, with $C_1> 0$ and $C_2\ge0$, for all $s\in\mathbb{R}$.
For me, it is true that $\int\limits_{-\infty}^\infty f(s)g(s) \, ds\le C_3<\infty$, with $C_3\ge0$. This statement would come from the fact that $f(s)g(s)\le C_2f(s)$, and then integrating on both sides.
This seems to be trivial, but I don't know if I should have additional conditions on the functions $f$ and $g$ so the previous statement is true. Any insights on this will be appreciated.
g(s) could be equal to a constant value per your statement then the integral would not be bounded. To be true g(s) integral must be bounded also.